♦ DIFFERENTIAL IDENTITIES OF PRIME RINGS WITH ......Key words and phrases. Differential identity,...

transactions of theamerican mathematical societyVolume 316, Number 1, November 1989

♦ DIFFERENTIAL IDENTITIES

OF PRIME RINGS WITH INVOLUTION

CHEN-LIAN CHUANG

Abstract. Main Theorem. Let R be a prime ring with involution * . Suppose

that 4>(xA,(x A)*) = 0 isa »-differential identity for R, where A; are distinct

regular words of derivations in a basis M with respect to a linear order < on

M. Then (j>(Zij,z*.) = 0 isa »-generalized identity for R, where z¡¡ are

distinct indeterminates.

Along with the Main Theorem above, we also prove the following:

Proposition 1. Suppose that * is of the second kind and that C is infinite. Then

R is special.

Proposition 2. Suppose that Sw(V) Ç R C LW(V). Then Q, the two-sidedquotient ring of R , is equal to Lw(V) ■

Proposition 3 (Density theorem). Suppose that DV and WD are dual spaces

with respect to the nondegenerate bilinear form ( , ). Let vx,..., vs, v [,...,

v's € V and ux,...,ut, u\,...,u\ G W be such that {vx,...,vs} is D-

independent in V and {«],...,«,} is D-independent in W. Then there exists

a e Sw(V) such that v¡a = v't (i = 1.s) and a*u¡ = u'. (j = l,...,t) if

and only if (v'.,Uj) = (v¡, «') for i = l,...,j and j = \,...,t.

Proposition 4. Suppose that R is a prime ring with involution * and that f

is a »-generalized polynomial. If f vanishes on a nonzero ideal of R , than f

vanishes on Q , the two-sided quotient ring of R .

The objective of this paper is to prove the following generalization of Khar-

chenko's theorem on differential identities to prime rings with involution (the

notation here is explained in §1 below):

Main Theorem. Let R be a prime ring with involution *. Suppose that

cf)(x¡ ', (x/)*) = 0 is a ^-differential identity for R, where A are distinct reg-

ular words of derivations in a basis M with respect to a linear order < on M.

Then cf>(z¡j,z*j) = 0 is a ^-generalized identity for R, where z(.. are distinct

indeterminates.

Received by the editors April 18, 1988.

1980 Mathematics Subject Classification (1985 Revision). Primary 16A38; Secondary 16A28,16A72, 16A12, 16A48.

Key words and phrases. Differential identity, generalized (polynomial) identity, left (Martindale)

quotient ring, two-sided (Martindale) quotient ring, prime rings with involution.

©1989 American Mathematical Society

0002-9947/89 $1.00+ $.25 per page

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252 CHEN-LIAN CHUANG

We have the following immediate

Corollary. Let R be a prime ring with involution *. Then any ^-differential

identity of R is a consequence of the basic identities (l)-(9) of §1 and *-

generalized identities of R.

There is a similar work [5] by Lanski. However the main result (Theorem

7) of [5] is false. Before proceeding, it seems proper to give the following

counterexample to Theorem 7 [5] (and Theorem 4 [5] as well):

Counterexample. (Using Lanski's type of notation [5].) Let Q be the field of

rational numbers and let F = Q(t), the field of rational functions in t with

coefficients in Q. Set 7? = Fn, the ring of all nxn matrices over F, and endow

7? with the transpose involution * ; that is, for x — (a.) E R, x* = (a..).

Define the derivation S on R by setting, for x = (a..) E R, x = ((d/dt)au),

where d/dt is the usual differentiation derivative with respect to /. Observe

that (x*)s = (xs)* for xeR.

Let etj (i,j = 1, ... ,n) be the matrix units of R ; that is, let et, be the

matrix of R with 1 in the (i,j) entry and zero elsewhere. Let

(I 0\

b = tien =i=i

2

VO nj

Define the derivation d on 7? by setting x - x +[b,x] for x e R. Now

consider s = xd + (x*) - [2b,x*] for x E R. We compute s — x +(x*) -

[2b ,x*] = xô + [b ,x] + (x*)s + [b ,x*]-2[b ,x*] = xs + [b ,x] + (xsf -[b ,x*] =

x +(x )* + [b,x—x*]. So 5 is always symmetric and hence exlse2x—ex2sexx =

0 . We have thus shown that

f(x,y) = ex|(xd + yd - [2b,y])e2x - eX2(xd + / - [2b,y])ex,

isa G*-DIof 7? in the sense of [5]. The derivation d is obviously outer and can

be chosen to be the first element in the well-ordering of the basis of the space

of derivations on 7? modulo the space of inner derivations (that is, M\M0

in [5]). Let 0 denote the empty set. Theorem 7 [5] asserts that f0(x,y)

= eX2[2b,y]exl - exx[2b,y]e2x is a G*-PI of 7?. But, since f0(ex2,e*x2) =

f0(eX2 ,e2x) = 2e2x jí 0, this is obviously false!

The crucial difference between Lanski's Theorem 7 [5] and our Main Theorem

above is as follows: In [5], * remains inside of derivation words. By introducing

d* for each derivation d on 7?, we are able to push * outside of the derivation

words in our Main Theorem and this gives the right form for doing induction.

Ignoring the falsity of [5], our improvement is to remove the multilinearity

assumption of [5]. Viewing the complexity of Kharchenko's analogous work

[4] for rings without involution, this part is quite intricate. For this work, a

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»-DIFFERENTIAL IDENTITIES OF PRIME RINGS 253

structure theory for *-primitive rings with socle is required and this is developed

in §11. There, as auxiliary results, we obtain an interesting characterization of

the Martindale two-sided quotient ring and we also prove the useful *-version

of the Jacobson density theorem.

One minor difference worth emphasizing is that the coefficients of our *-

differential identities must be allowed to lie in the left Martindale quotient ring

of R, instead of the two-sided Martindale quotient ring as in Lanski [5]. This

is not simply a matter of generality and is actually crucial to our proof, as will

be seen in §IV.

Our method is by mixing Kharchenko's techniques with a theorem due to

Rowen. The material is organized as follows: In §1, we generalize Kharchenko's

original theorem slightly and meanwhile explain our notations. In §11, we treat

the case when * is of the second kind and introduce the notion of special

differential identities. In §111, we generalize those well-known results on rings

with nonzero socle to their *-versions. The proof of the Main Theorem is given

in §IV.

Along with proving the Main Theorem above, the interesting auxiliary results

obtained are singled out under the title "Proposition".

Most of the notation here is adopted from Kharchenko's original work [3, 4]

instead of Lanski [5].

I. Preliminaries

Suppose that 7? is a prime ring and that F is the set of all its nonzero

ideals. Let RF be the ring of left quotients of 7? relative to F ; that is, 7?^ =

lim Hom(Ä7,7?). Consider the subring Q of RF consisting of those elements

a E RF for which there exists a nonzero ideal Ia E F such that ala Ç 7?. Q is

called the two-sided quotient ring of 7? relative to F . The center of Q, denoted

by C, coincides with the center of RF and is called the extended centroid of

R.

Let Der Q consist of all derivations on Q. For o E Der Q and a e C,

define aa by xaa = (xa)a for x E Q. For a,p. E T>erQ, define x -

(xa Y - (xM)a for x E Q. o a and [o ,p] thus defined are also derivations

of Q. In this manner, Der Q is a right vector space over C and is also a

Lie algebra (over the subfield consisting of a E C such that a" — 0 for all

a E Der Q) . Der Q is called the differential Lie C-algebra of derivations in Q .

Let Der 7? consist of all derivations on R and let D = (Der 7?) • C. Obviously

D C Der Q .

By a differential polynomial, we mean a generalized polynomial involving

noncommutative indeterminates which are acted by derivations of 7? as unary

operations. We allow the coefficients of a differential polynomial to lie in RF .

Obviously, every differential polynomial can be written in the form cf>(xjJ),

where <p(zi..) is an ordinary generalized polynomial in z( and A are words

of derivations of 7?. <f> = 0 is said to be a differential identity for R if

cfi assumes the constant value 0 for any assignment of values from R to its

indeterminates.

The following basic differential identities hold in any prime rings:

( 1 ) (xy)a = xay + xya , where o E Der Q.

(2) (x + y)a — xa + ya , where o E Der Q.

(3) xa = xa - ax, where a is the inner derivation defined by a E Q.

(4) x[<T""' = (x")ft - (xß)° , where a,pE DerQ and [o,p] is their commu-

tator.p-times

,-*-,

(5) (■ ■ • ((x ")") ■■■)" = x , where o e DerQ and where p is the char-

acteristic of the given ring. If p = 0, then this identity assumes the form

x = x.

(6) xaa+"P = (xa )a + (xß)ß , where a , p E Der Q and a,ß EC.

Let Z)jm be the Lie subalgebra of Der Q consisting of all inner derivations

of Q . We choose a basis MQ for the right C-vector space 7>m and augment

it to a basis M of 7>m + D. We also fix a total order > in the set M such

that p0 > p for pQ E MQ and p E M\MQ. We then extend this order to the

set of all derivation words in M by assuming that a longer word is greater than

a shorter one and that words of the same length are ordered lexicographically.

By a regular word in M , we mean a word of the form A = ôx'ô2 ■ ■ ■ ôs™ such

that

(1) Ó:EM\M0 for /= 1, ... ,m,

(2) ôx < ô2 < ■ ■ < ôm , and

(3) s■ 0.

By means of the basic identities (l)-(6), any differential identity can be trans-

formed into the form <p(xi ' ) = 0 where

(1) <p(zr) is a generalized polynomial in distinct indeterminates z..,

and where

(2) A are regular words in M .

Kharchenko [4] has proved the following (actually for semiprime rings with

characteristic):

Theorem (Kharchenko). Let R be a prime ring. Suppose that cp(x¡') = 0 is a

differential identity for R, where A are distinct regular words. Then </>(z/y) = 0

is a generalized identity for R, where z, are distinct indeterminates.

Now suppose that R is a prime ring endowed with the involution *. Note

that * can be uniquely extended to Q in the following manner: Let a E Q

and let I E F be such that la ç R and al ç R. Define a* by the following:

a* : RI* —> RR via r*a* = (ar)* . It is obvious that a* E RF . We can verify

easily that for r E I, a*r* = (raf E R. So a* E Q as desired. We assume

henceforth that * is defined on the whole Q. We say that * is of the first kind

if a* = a for all a E C . Otherwise, we say that * is of the second kind.

»DIFFERENTIAL IDENTITIES OF PRIME RINGS 255

By a ^-differential polynomial, we mean a generalized polynomial involving

noncommutative indeterminates which are acted by the involution * as well as

derivations of 7?, regarded as unary operations. Although, in general, * cannot

be extended to RF , we still allow the coefficients of a ^-differential polynomial

to lie in RF . The following is an example of a *-differential polynomial:

4>(x,y) = axx* '*a2y 2*a3 + bxyb2y 2x*b3,

where ax, a2, a3, bx, b2, b3 E RF and ôx, ô2 E 7>m + D.

For a *-differential polynomial cf>, we say that cf> — 0 is a ^-differential

identity for R if d> assumes the constant value 0 for any assignment of values

from R to its indeterminates.

The following basic identities are trivial for any rings with involution * :

(7) (xy)* = y*x*.

(8) (x + y)* = x* + y*.

For er e Der Q, we define o* as follows: For x E Q, x = ((x*)a)*.

It is easy to verify that a*, thus defined, is also a derivation of Q. The

following basic *-differential identity holds in any rings with involution * :

(9) (xaf = (x*){a'], where aeDerß.

An immediate generalization of (9) is the following:

(9)' (xSv"Sn)* = (jc'jWW-W), where sx,ô2,... ,8n E DerQ.

By means of (9)', we can push * inward or outward through derivations.

Bringing * outside of derivations, every »-differential identity can be trans-

formed to the form <f>(x¡ ', (xiA)*) = 0, where A are words of derivations

only, not containing any *. By using identities (l)-(8), we can further trans-

form each A into regular words as we did in Kharchenko's theorem. From

now on, unless specified otherwise, whenever we write cp(x¡ ', (xtJ)*), A are

always understood to be regular words.

Our main objective is to prove the following »-version of Kharchenko's the-

orem:

Main Theorem. Let R be a prime ring with involution *. Suppose that

(p(x¡■' ,(x¡ J)*) = 0 is a »-differential identity for R, where A are distinct reg-

ular words of derivations in M with respect to <. Then </>(z.. ,z*.) = 0 is a

^-generalized identity for R, where z.. are distinct indeterminates.

Now we prove an immediate generalization of Kharchenko's theorem, which

will be needed later. Suppose that for each indeterminate x-, we pick a basis

M{0'} for DiM and then augment MlQ'] to a basis M{,) for DiM + D. We also

fix a total order <(,) in the set M{l) such that p <(,> p0 for p0 e M^ and

p E M{i)\Mq} . We then extend this total order to the set of words in M(>)

and also define regular words in M(l) with respect to <(,), as we did before

for M with respect to < when we stated Kharchenko's theorem. If desired,

M(l), < can be made to coincide with each other. But in general, they are

completely arbitrary and independent of each other.

By means of the basic identities (l)-(6), any differential identity (withouta'"

involution *) can be brought to the form <p(xi ' ) - 0, where

(1) cp(ztj) is a generalized polynomial in distinct indeterminates z..,

and where

(2) for each i, A{l) are regular words in tV7(,) .

We have the following:

Generalized Kharchenko theorem. Let R be a prime ring. Suppose that

4>(Xj ' ) = 0 is a differential identity for R, where Av' are distinct regular words

in Mi with respect to <(i) for each i, j. Then cf>(zij) is a generalized identity

of R, where z are distinct indeterminates.

Proof. Let us assign arbitrarily certain fixed values from 7? to the indetermi-(1) A(l'

nates x2,x3, ... of cp other than xx . The resulting expression dy- '(x > ) = 0

is a differential identity involving the indeterminate xx only. Applying the orig-

inal Kharchenko's theorem to </5 = 0, we obtain that cfy (zXj) = 0 is a gener-

alized identity for R . Since the assignment of values from R to x2,x3, ... is

A<"completely arbitrary, <p(zXj ,x¿ ' )¡>2 = 0, the identity obtained by substituting

A(l» AUI

Zj. for xx' in 4>(x¡' ) = 0, is still a differential identity for 7?. ContinuingA(/)

this process, we can eventually replace all x¡ ' by x( . This completes the

proof.

If we let M{1), <(,) all coincide with M = M{X), <=<(l), then the above

generalized theorem specializes to the original one.

Back to the »-version, let M0, M, < be as explained in Kharchenko's

original theorem. Define tV/J = {p* : p E M0} and M* = {p* : p E M) . For

px , p2 E M, we define p* <* p2 if and only if px < p2. For a word of

derivations A = 5X62■ ■ ■ ôn , where ôt E M, we define A* = f5*f52 • ■ -S*. Note

that if A is a regular word in M with respect to < , then A* is a regular word

in M* with respect to <*.

Using the identity (9)', cp(xfi, (xt J)*) = 0, where A. are regular words in

M, can be transformed into cp(xf' ,(x*) ' ) = 0, where A. and A* are regular

words in M and M *, respectively.

II. Special »-differential identities

Let y/(xf' ,(x*)áj) = 0 be a »-differential identity for R, where A., A^

are simply words of arbitrary derivations, not necessarily regular words in M.

Then y/(xf' ,(x*)&J) = 0 is said to be special for 7? if y/(x¡ 'Ay A) = 0 is

a differential identity for 7?, where y¡ are new indeterminates distinct from

x¡. In other words, y/(xfJ ,(x*)A') = 0 is special for 7? if x¡ and x* can

be regarded as independent indeterminates. This generalizes the definition of

special generalized identities given in [7, p. 471].

Suppose that we use identities (l)-(6) to transform y/(xf', (x*)A') = 0 into

another identity d(xfJ ,(x*)A>) = 0, where A., A. are also words of arbi-

trary derivations (not necessarily regular in M ). Then the same procedure also

transforms y/(xfJ ,yf'J). = 0 into 6{xfJ ,yfJ)=0. Thus if y/(xfJ ,(x*f>) = 0

is special for R, then so is 6(xA ,(x*) ') = 0. We have thus shown that

the speciality of a ^-differential identity is independent of transformations using

identities (l)-(6).

If every »-differential identity of the form y/(xA', (x*)Aj) = 0, where A.,

A' are words of arbitrary derivations (not necessarily regular), is special for 7?,

then 7? is said to be special. The relationship between the speciality and our

Main Theorem is the following:

Lemma 1.7/7? is special, then the Main Theorem holds.

Proof. Suppose that <f>(xA ,(xA)*) = 0 is a »-differential identity, where A

are regular words in M. Using the identity (9)', <¡>(xA ,(xf')*) = 0 can be

transformed into cp(x¡ ', (x*) ' ) = 0, where A* are hence regular words in M*.

Suppose that R is special. Then <p(xi' ,yt■') - 0 is a differential identity for

7?, where y¡ are new indeterminates. By the generalized Kharchenko theorem,

(p(z¡j, w¡ .) = 0 is a generalized identity for R. In particular, <p(z¡ , z*.) — 0 is

a »-generalized identity for 7?.

The aim of this section is to prove the following generalization of Theorem

7 [7, p. 473]:

Proposition 1. Suppose that * is of the second kind and that C is infinite. Then

R is special.

The proof will be completed by a series of lemmas. First, we recall some more

definitions from [3]. Let Q' be the ring anti-isomorphic to Q with the same

additive group. That is, Q' is the opposite of Q. Let B denote the subring

of the tensor product Q <g>z Q (where Z is the ring of integers) generated by

the elements of the form 1 ® r, rigil, r e R. Elements of B are of the form

J2¡ r ® vi, where r., vt e R. For a E Q, ß = £, r¡ ®v¡ e Q ®z Ql , let

a • ß = ¿/ «W • *™ a E Q, let aL = {ß E B: a ■ ß = 0} . For V ç B, let

V = {a E RF : a • V = 0}. If A is an endomorphism of the abelian group

Q, we set ß = £\ r¡ ® vj. (This is well defined!) If f(x) is some linear

expression involving the variable x and if ß = J2, r¡ ®vtEB, then we set

f(x) • ß = Y,j Vjf(rjx) ■ Observe that if f(x) is identically zero on R, then so

is f{x) ■ ß .

We quote the following [3, Lemma 1]:

Lemma 2. Suppose that ai e RF (i = 1,2, ... , m). Then

777 / 77! \ -1-

Z"iC=(n«t) ■7=1 \l=l /

Suppose that cp(xj ', (xt ')*) = 0 is a »-differential identity, where A are reg-

ular words in M. cp(x¡ ', (xt')*) = 0 is said to be trivial if the »-generalized

polynomial <f>(zi., z*.) is trivial, or equivalently if the ordinary generalized poly-

nomial ^(z,.. ,j>..) (without *) is trivial.

A linear »-generalized identity f(x) is of the form

fix) = J2 aixbi + Ç cjx*dj = °.' j

where ai,bi,Cj ,d- E RF . f is trivial if and only if J2¡ a¡ ®c bi ~ ® and

X),- Cj ®c d= 0. A linear »-differential identity cfi(x) can be written in the

form

77, 777,

cp = d>(x\(xrr)=ET.a.k]xA,b,k)+EEc?^r;)*<).i k=l j /=1

where a¡ ,b\A ,c. ,d E RF and where A;, T are regular words in M.

Using the identity (9)', cp can also be written in the form

71, 777,

cp = 4>(x\(x*f>) = Y.T,4k)*V + ££c;V)r;<>,7 k=\ j 1=1

where V* are, certainly, regular words in M*. cp is said to be trivial if and

only if E"'=i a[,k) ®c bf] = ° for a11 ' and ^/=i ci" ®c df = ° for a11 J ■The following lemma is essentially the »-version of Lemma 2 [3].

Lemma 3. Suppose that every linear »-generalized identity of R is trivial. Then

so is every linear »-differential identity of R .

Proof. We need the following formula from [3, p. 158]: Let A = Sx ■ ■ âm be a

regular word in M such that Sx = S2 — ■■■ = Ss ^ Ss+X , where 0 < s 0. Let ß E B . Then

(1) (axAb) -ß = (a- ß)xAb + s(a ■ ßS')x'h"amb + ■■■ ,

where the dots above denote a sum of terms dx b in which A < S2 ■ ■ ■ ôm .

We define a partial order on ordered pairs of regular words in M . Assume

that A! , A,, r. , and T-, are regular words in M. We define (A,, I",) >

(A2, r,) if A, > A2 and Tx > T., or if A, > A., and Tx >T2.

Let M be a finite subset of M. Note that the set of all regular words in

M is well ordered under < . Hence the partial order < on word pairs defined

above is well-founded when restricted to ordered pairs of regular words in M .

Let J^(M) denote the set of all linear »-differential identities involving only

derivation words in M. J?(M) consists of the »-differential identities of the

form

££^X + ££^V^7 = oi k j I

where A., T are regular words in M.

Now suppose that

77, m j

m = £Efl/V*!fc)+££4V0H(/) = oi k=\ 7 l=\

is an arbitrary linear »-differential identity in S(M) ; that is, A(., T are

regular words in M. We may assume that (A,,Yx) is the leading word pair of

cp ; that is, A, > A. and Yx > Y for all i, j . We proceed by induction on the

leading word pair (A! ,YX) (under the partial order defined on word pairs) to

show that cp is trivial.

If (A,,r,) = (0,0), then <f> is simply an ordinary »-generalized identity

without derivations. By our assumption, cp must be trivial. So suppose that

(A1,r,)/(0,0). Let us first assume that A,/0. Obviously a\X), ... ,a\"']

can be assumed to be C-linearly independent.

We claim that it suffices to prove the case when nx = 1 : Suppose that nx > 1

is given. Let ß E f]"k'>2{a\ ) • Using formula (1), cp(x) • ß has a single term

(a\ ] ■ ß)x lb\ containing the regular word A, . If we have proved the case

when nx = 1, then cp(x) • ß is trivial and hence a\ ' • ß — 0. So a\ • ß — 0

for all ß E fl^^S ))± and' by Lemma 2, a[X) E J2l>->a[ 'C, a contradiction

to the C-linear independence of a\ , ... ,a" .

So we may assume nx — I . Suppose Ax = SXS2 ■■■ S , where Sx = S2 — ■• ■ =

ôs / ös+x and 0 < s 0. Set A, = ô2 ■ ■ ■ Sm . Let px, ... ,pn

be all derivations in M other than <5, such that /j,A, , ... ,/i„A, occur in cp

as some A(. Let us say Ar = pxAx, ... ,Ar = pnAx . We also assume that

A, occurs in cp as Ak . In view of formula (1), the sum of terms of cp(x) • ß

containing x ' has the following form:

71 71 r,

,(fc) 0/7twA,A(7t) , X^fji) öwA,,(i)9k„

= 1 tV=1 i

s(a\X) • ßs')xA'b\X) + E£(<' • ß*)M? + £(<' • ß)^bl

If a\ • ß — 0, then the leading word pair of cp(x) • ß is strictly less than that

of cp. Since cp(.x) ■ ß E S(M), by our induction hypothesis, <p(x) ■ ß must be

trivial. So we have

(2) s(a\l) ■ ßd>)®c b\X) + J2K] ■ ß"') ®c K, + £(< ■ ß) ®c Kl = 0-i .k i

Since 0 < s 0, 5 is nonzero as an element of

C. Hence a[X) ■ ß ' can be linearly expressed in terms of a^ ■ ßM* and a{k ■ ß .

Using the linearity of ( ) • ßßx, () • ß , and introducing new notation, we have

a[X).ßs>+^2dT.ßß<+h.ß = 0.T

Since the right coefficients of (2) do not depend on ß, neither do dr and h .

This shows that the mapping

C:a[X).ß^a\X).ßai+J2dT-ßßT+h-ß,X

where ß ranges over B, is well defined. Choose I E F such that Ia\ ç R,

IdTCR (t= 1, ...,«), and Ih ÇR. Set T = B(l®I). If ß E T = B(I®I),

then both a[x) ■ ß and Ç(a[x) ■ ß) fall in 7?. Thus C is defined on the ideal

{a\ • ß : ß E T) of R and the range of this ideal under Ç is also contained in

R. Furthermore, for v E R,

C(v(a[X)-ß)) = C(a[X)-ß(l®v))

= a[X] ■ ßaA[l ® v) + J2dz ■ ß"1 (I ® v) + h ■ ß(l ® v)T

= vC(a{X,-ß).

Hence, by the definition of RF , there exists t E RF such that, for ß E T,

(a{xX).ß)t = aa\l)-ß) = a\X).ßSi+J2äz-ß'i' + h-ß.T

For x E R , we compute

(a[X) ■ ß)(xt) = ((a\X) ■ ß)x)t = (a\X) ■ ß(x ® l))t

= a(X) ■ (ß(x ® l)f +Y,dx- (ß(x ® O)"' + h ■ (ß(x ® 1))T

= (a\X).ßa>)x + (a[X).ß)xS'

+ £(^r • ßßlx + £(^r • ß)*"' + (h ■ ß)x,X X

and

(a[x) ■ ß)(tx) = ((a\X) ■ ß)t)x = (a\X) ■ ßä')x + £>r ■ ß>")x + (h ■ ß)x.X

Hence we obtain

(3) (a1/' • ß)[x, t] = (a[X) ■ ß)(xt - tx) = (a\X) ■ ß)xSi + ^(^r ■ ß)xM'.X

Now suppose that a[X) — ax,a2, ... is a basis for the subspace a] ■ C +

J2 d ■ C . Express d in terms of this basis:

dx = axax + ■ ■ ■ , ax E C.

Let ß E n/>2 afnT. Then dT-ß = aT(ax ■ ß). By (3), we have

(a[X) .ß)([x,t]-xS> -J2*xxß') =Q-

Hence [x, t] = x ' + J2r otxx^ . This contradicts with our choice of the basis

M. So we must assume that A, = 0 and Yx ̂ 0 .

Replacing x by x* in <j> and using the identity (9)' to bring * inside of

derivations, we have

71/ mi

I 7C=1 7 /=1

By a similar induction on regular words in M — {ô* : ô E M} with respect to

< , we can show that Y. — 0 . This is another contradiction. We have thus

shown that any linear »-differential identity of J^(M) is trivial.

Now let

*(*> -Et^A'bi}+££c,VT^ = o7 7C=1 7 1=1

be an arbitrary linear »-differential identity. Let M be the finite subset of M

consisting of all derivations occurring in A, and Y.. Then cp e J?(M). By the

result of the previous paragraph, cp must be trivial.

Lemma 4. Suppose that every linear »-generalized identity of R is trivial. If

cp(xA ,(xA)*) = 0, where A are distinct regular words in M, is a multilinear

^-differential identity for R, then cp(zj.,yij) = 0, where z.., y¡¡ are distinct

indeterminates, is a multilinear generalized identity of R.

Proof. If cp involves only one indeterminate, say xx , then cp = cp(xx ', (xx ')*) is

trivial by Lemma 3. Hence cp(zXj ,yXj) = 0 is also a trivial generalized identity

for 7?.

Now suppose that cp involves more than one indeterminates, say xx, ... ,x .

Let us assign certain fixed values from R to x2, ... ,x . The resulting identity

thus involves the indeterminate xx only and hence must be trivial by Lemma

3. As in the previous paragraph, we can replace xxJ , (xx')* by new indeter-

minates Zj ■, yXj to obtain a generalized identity. Since the values assigned to

x2, ... ,xn are completely arbitrary, cp(zXj ,yXj ,xf' ,(xf')*)(>2 = 0, the iden-

tity obtained by replacing xx' , yx' in cp = 0 by z. , yx , respectively, also

holds on R. Repeating the same argument for x2, ... ,xn, the desired result

follows.

By assigning fixed values to all indeterminates but one, we can reduce a given

identity to an identity involving only one indeterminate. This technique, used

in proving the generalized Kharchenko theorem and also in the proof above,

will be used frequently throughout this paper.

Lemma 5. Suppose that is of the second kind. Then every multilinear

^-differential identity of R is special.

Proof. By Theorem 7 [8, p. 473], every multilinear »-generalized identity of R

is special. In particular, every linear »-generalized identity is special and hence

must be trivial by Lemma 1.3.2 [1, p. 22]. The result now follows from Lemma

4.

Lemma 6. Suppose that * is of the second kind and that ch 7? = 0. Then R is

special.

Proof. Let cp be a given »-differential identity. Replacing each indeterminate

x( by 2x;, 3x;, ... and using the Vandermonde determinant, we can solve

the homogeneous components of cp. Hence each homogeneous component of

cp is also a »-differential identity of R. It suffices to show that each homo-

geneous component of cp is special. Replacing cp by one of its homogeneous

components, we may assume from the start that cp is homogeneous in each

indeterminate.

Let 6 be the multilinearization of cp. By Lemma 5, 6 is special. Identifying

the indeterminates in 6 properly, we obtain ncp, where « is a nonzero integer.

Hence ncp is also special and so is cp, since chT? = 0.

Lemma 7. Suppose that * is of the second kind and that ch R = p > 0. If C is

infinite, then R is special.

Proof. By the technique used in proving Lemma 4 (and also the remark fol-

lowing), it suffices to prove the case when cp involves only one indeterminate,

say x. We proceed by induction on the degree of cp. If deg 1 . As the

induction hypothesis, we also assume that the result holds for any »-differential

identity with less degree. Bringing * inside of derivations by identity (9)' and

suppressing words of derivations for simplicity of notation, we write cp(x ,x*)

for cp(x ', (x*) '). Now consider

0(x,x*,j;,/) = cp(x + y,x* +y*) - cp(x,x*) - cp(y ,y*).

By the induction hypothesis (and the remark following Lemma 4), 6 is special

for R . Now regard x , x*, y , and y* in 6 as independent indeterminates as

granted. Set x = x, x* = 0, y = 0, and y* = x* in 6 :

6(x,0,0,x*) = (p(x,x*) - <p(x,0) - (p(0,x*).

Thus cp(x,0) + cp(0,x*) is also a »-differential identity for R. We also have

the identity 0(x,x*) = 6(x,0,0,x*) + cp(x,0) + cp(0,x*). Since 6(x,0,0,x*)

is special for 7?, it suffices to prove that cp(x,0) + 0(0, x*) is special. This is

equivalent to showing that both <p(x,0) = 0 and 0(0,x*) = 0 are differential

identities for 7?.

Set Ç(x) = 0(x,O) and r¡(x*) = 0(0,x*). For positive integers t > 0, let

C,(x) and t].(x*) denote the sums of those monomials with degree t in Ç(x)

and in t](x*), respectively. Pick sufficiently but finitely many a e C such that

a* = a . Let 7 be a nonzero »-ideal of 7? such that apI ç R for those finitely

many a E C we have picked. Then Ç(apx) + rj(apx*) vanishes on 7. Since

ap are constants of any derivations, we can use the Vandermonde determinant

to solve the homogeneous components of C(c/x) + t\(apx*) = 0. So we have

that, for each t > 0, Ç,(x) + t]t(x*) vanishes on 7.

We claim that there exists a E C such that apt ^ (a*)pl : Pick ß e C such

that ß* ± ß . For s E C such that s* = s, set a(s) = s + ß . Observe that

a(s)* =s + ß* =u ß'-ßa(s) s + ß s + ß

Hence if s ^ s', then a(s)*/a(s) ^ a(s')*/a(s'). Since the equation ypt = 1

has only finitely many solutions in the infinite field C, there exists a symmetric

element seC such that a(sf ¿ (a(s)*)pl.

Pick a E C such that apl ^ (a*)pl as claimed above. Let 7 be a nonzero

»-ideal of R such that aJ ç 7?. Then Çt(apx) + nt((apx)*) = 0 holds for

xeIJ . Hence for x € 77 , 0 = ap'(Ct(x) + t]t(x*)) - (Çt(a"x) + r¡t((a"x)*)) =

(apl - (a*)pl)r]t(x*). Thus r¡t(x') = 0 holds on 77. Similarly, £;(x) = 0

also holds on 77. Note that Çt(x) = 0 and r\t(x) = 0 are simply ordinary

differential identities without the involution *. By the remark on p. 74 in [4],

Ct(x) = 0 and ^(x) = 0 also hold on R. This completes the proof.

Proposition 1 now follows from Lemma 6 and Lemma 7.

One might wonder whether or not the assumption that C is infinite can be

eliminated in Proposition 1. The answer is negative, as the following example

shows.

Example. Fix a positive integer n > 1 . Let 0 with p " elements. For a E O, define a — ap . - is an involution of

the second kind on O. The symmetric elements of satisfy the polynomial

identity s = s. But the »-identity (x + x) = x -l- x is obviously not special.

For a noncommutative example, we can take any k x k matrix ring k

(k > 1) with transpose involution * defined as follows: For a¡¡ GO (i,j =

1, ... ,k), (a¡j)* - (ccjj). For an infinite-dimensional example, let V be an

infinite-dimensional vector space over O. Pick a Hermitian bilinear form ( , )

on 0F. That is, ( , ) satisfies the condition iav ,ßw) = aß(v ,w) for a,ßE

<t> and v ,w E V . Our desired ring is the set consisting of all continuous finite

rank linear maps in End^F). (See §111 for definitions.)

Actually, we can prove that any counterexample to Proposition 1 must be of

the form described above.

III. »-RINGS HAVING MINIMAL ONE-SIDED IDEALS

We digress here to generalize some basic theorems on rings having minimal

one-sided ideals to their »-versions.

First, we recall some basic definitions. Let D be a division ring. Suppose

that D V is a left vector space over D and IVD , a right vector space over D.

If there exists a nondegenerate bilinear form ( , ) : V x IV ^ D, then D V and

IVD are said to be dual with respect to ( , ).

Let A(V) = Hom(DV,DV) and A(W) = Hom(WD,IVD). For a E A(V)

and b E A(W), if (va,w) — (v ,bw) holds for all v E V and for all w E W,

then we call b an adjoint of a . Not every element a E A(V) has an adjoint.

But if a G A(V) does have an adjoint, then that adjoint must be unique and

will be denoted by a*. a E A(V) is said to be continuous if a has an adjoint

a* E A(1V). The set of all continuous elements a E A(V) forms a ring and

is denoted by LW(V). Let SW(V) = {a E LW(V): dimD(Ka) < 00}. For

u,, ... ,vn E V and wl,... ,w E IV, the linear transformation a E A(V)

defined by va = (v ,wx)vx + ■ ■ ■ + (v ,wn)vn is in SW(V) and has the adjoint

a* E S y (IV) given by a*w — wx(v x ,w ) + ■■• + wn(vn ,w). Conversely, any

a E SW(V) can be represented in the form described above. The following two

basic facts can be found in [2].

Fact 1. Let vx, ... ,vn E V be 7>independent. Then there exist D-indepen-

dent vectors ux, ... ,unE IV such that (il , m .) = ôt.for /, j — 1,2, ... , n .

Fact 2. Let VQ and IVQ be finite-dimensional subspaces of D V and H^ , respec-

tively. Then there exist finite-dimensional subspaces Vx 2 VQ and Wx D IV0 of

D V and IVD respectively such that the bilinear form ( , ) restricted to Vx x Wx

is nondegenerate. (Hence dim^ = dimH^ by Fact 1.)

For a prime ring R, let Soc(7<) be the sum of all minimal right ideals of

7? . It is well known that Soc(7?) is also the sum of all minimal left ideals of R

and that Soc(7?), if nonzero, is the unique minimal ideal of 7? . We have the

following representation theorem for prime rings with nonzero socle.

Jacobson theorem. Let D V and IVD be dual spaces over a division ring D with

respect to the nondegenerate bilinear form ( , ). Let R be a subring of A(V)

such that SW(V) ç R ç LW(V). Then R is a primitive ring with Soc(7?) =

SW(V).

Conversely, given a prime ring R with Soc(7?) / 0, we can find a division

ring D and a pair of dual spaces DV and IVD such that SW(V) ç R c LW(V).

The following fact can be found in [4, p. 171] or Proposition 7, [6, p. 98].

However, we give here a new proof using the density theorem, which is more

related to our next proposition.

Fact 3. Suppose that SW(V) CRC LW(V). Then RF = A(V).

Proof. Let h e RF . Since SW(V) is the unique minimal ideal of R,

Sw(V)h ç R. We define h E A(V) as follows: For v¡ E V and ri E SwiV),

h: Y,viri ^ 12vi(rih). Note tnat r,b G R and hence vf.rfi) E V. We

must show that h is well defined: Suppose J2vjr¡ - 0- Since r g SW(V),

dimD(X); Vr/) < oo. Choose r e Sw(V) such that r restricted to X,, Vr¡ is

the identity map. Then r¡r = r¡. Hence

5>/W = !>2vttrir)h) = T.viWrh)) = £¥()('*) = °-

Via this natural embedding /i —► h , we have 7<F ç /1(F).

Conversely, suppose that h E A(V). In order to prove that h E RF, we

must show that Sw(V)h ç R : Every element a G S^K) can be represented

by the form va = ^2¡(v ,w¡)v¡., where v ,v¡ E V, w¡ G IV. Hence vah =

J2]iv ,u;(.)(t;(./z). Then a/2 g SwiV) E R as desired.

The following proposition is interesting in itself.

Proposition 2. Suppose that SwiV) CRC LW(V). Then Q = LwiV), where

Q is the two-sided quotient ring of R.

Proof. Suppose that h E LwiV). Then hSw(V) C SW(V) ç R and Sw(V)h çSW(V)CR. So hEQ.

Conversely, since Q ç R , by Fact 3 above, we may assume QcA(V). Let

hEQ. In order to show that h E LW(V), we must find its adjoint h* E A(W).

Let SW(V)* = {r*:rESw(V)}, R* = {r* : r E R} , and LW(V)* = {r* : r E

LW(V)}. Observe that SW(V)* = SV(IV) and LW(V)* = LV(W). Hence

SV(W) ER* E LV(W). Let F* be the filter of nonzero two-sided ideals of R*

and let F.R* be the right quotient ring of R* relative to the filter F*. By a

symmetrical version of Fact 3 for right quotient rings, we have F.R* — A(W).

Given h E Q, we define h* : SV(1V) —► R* as follows: For t E SW(V), set

h*(t*) = (ht)*. Since h E Q, we have hSw(V) ç R as well as Sw(V)h ç R.

So ht E R and hence (ht)* exists. Thus h* is well defined.

For r E R and t E SW(V), h*(t*r*) = h*((tr)*) = (h(tr))* = ((ht)r)* =

(ht)*r* = h*(t*)r*. So h* is a right 7?*-homomorphism and hence h* EF, R* =

A(W). To prove that h* is the adjoint of h , we must verify that (v , h*w) =

(vh, w) for any v e V , w e IV.

Using the fact that SV(IV) = SW(V)* and the Jacobson density theorem,

there exists / G SW(V) such that t*w = w . Then we compute

(v ,h*w) = (v , h*(t*w)) = (v ,(h*t*)w) = (v ,(ht)*w)

= (v(ht),w) = ((vh)t,w) = (vh,t*w) = (vh,w).

So h* is really the adjoint of h and hence h E LW(V) as desired.

The following generalization of Theorem 7.3.16 [9, p. 269] is the »-version

of the Jacobson density theorem.

Proposition 3. Suppose that D V and IVD are dual spaces with respect to the

nondegenerate bilinear form ( ,). Let v,,..., vs, v'x,...,v's e V, and

ux, ... ,ut, u'x, ... ,u't E IV be such that {vx , ... ,i\} is D-independent in

V and {m, ,... ,u,} is D-independent in IV. Then there exists a E SW(V)

such that v¡a = v\ (i = I, ... ,s) and a*u — u'¡ (j = I, ... ,t) if and only if

(v'i. Uj) = (vj,u'J) for i = I, ... ,s and j = I, ... ,t.

Proof. The necessity is easy: Suppose that there exists a G SW(V) such that

vta = v'j (i = I, ... ,s) and a*u. - u. (j = I, ... ,t). Then

(v'i, Uj) = (v(a, Uj) = (v¡,a*Uj) = (v¡, u'ß.

To prove the sufficiency, by Fact 2, there exist finite-dimensional subspaces

VQ, IVQ of V, IV respectively such that vl,...,vs, v'x,...,v's e Vq and

ux, ... ,ut, u'x, ... ,u't E IVQ and such that the bilinear form ( , ) restricted

to VQ x IVQ is nondegenerate. Also VQ and IVQ are of the same dimension, say

of dimension n . Augment {vx, ... ,vs} to a basis {vx, ... ,vs, vs+x, ... ,vn)

for V0.

By Fact 1, there exist x,, ... ,xt E VQ such that (x¡,u¡) = <5( for i,j =

1, ... ,t. For each s < k < n , define v'k = Y^'/=i(vk > u'/)xi ■ Then for s < k < n

and for j — I, ... ,t,

K'w,)= ( ¿K »"/)■«/»MyJ =iVk,u'j).

Together with the originally given v[, ... , v's, we have that

(v., u ) = (v¡ ,u'j) for i - I, ... ,n and for j - I, ... ,t.

By Fact 1 again, pick a basis {wx, ... ,wn} for IVQ such that (<7.,w.) =

S¡j for i,j = 1, ... ,n. Now we define our desired a G S[V(V) by va =

2~2"=\(v >wj)v'j ■ Observe that v¡a = (v^w^v^ = v\ (i = I, ... ,n). So it

suffices to prove that a* Uj = «' for j = 1,...,/.

Given Uj (1 < j < t), compute (w(. ,a*w.) = (v¡a,Uj) = (v'Ûj) for / =

1, ... ,n . Since v(' (/ = 1, ... ,n) are chosen so that (v't,Uj) = (v¡,u'j) for

j = 1,...,/, we have (il ,a*^) = (w(', uß = (v¡, uß . So (v¡,a*u} - u'ß = 0

for /' = I, ... ,n. Hence (VQ,a*u. - u'ß = 0. Note that a* is defined by

a*u = ¿Z,"=i wi(v'i>u) € ô ôr any ueW . So, in particular, a*«. G W^ and

hence also a* m - «'. G IV0 . Since the bilinear form ( , ) restricted to VQ x IVQ

is nondegenerate, we have a* u. - u. = 0 as desired.

Let D be a division ring endowed with involution * and let DV be a left

vector space over D. V can also be regarded as a right vector space by the right

scalar multiplication defined by va = a*v , where v E V and q G D. Note

that A(DV) = A(VD). For this reason, the action of a G A(VD) on VD will also

be written on the right-hand side. Suppose that there exists a nondegenerate

bilinear form on D V x VD . Then D V is said to be a self-dual space with respect

to ( , ).


The »-version of the Jacobson theorem is the following well-known:

Kaplansky theorem [ 1, p. 17]. Let R be a prime ring with involution * and with

Soc(Tv) t¿ 0. Then there exists a vector space V over a division ring D endowed

with involution, which is self-dual with respect to a Hermitian or alternate bilinear

form ( , ) in such a way that

1. SV(V)CRCLV(V),

2. the * of R is the adjoint of R relative to this bilinear form ( ,).

Furthermore, the Hermitian and alternate cases are mutually exclusive, occur-

ring according as there is, or is not, a minimal symmetric idempotent e = e =*

e .

Conversely, any R such that SV(V) ç R ç LV(V) and such that R* = R

must be a prime ring with involution * and with Soc(Tv) = SV(V) ^ 0.

Proposition 4. Suppose that R is a prime ring with involution * and that f

is a ^-generalized polynomial. If f vanishes on a nonzero ideal of R, then f

vanishes on Q, the two-sided quotient ring of R.

Proof. Suppose that / vanishes on a nonzero ideal I of R. Replacing 7 by

7 n 7*, we may assume from the start that 7 is a »-ideal of R. We may also

assume that / is nontrivial. By Theorem 9 [8, p. 477], 7 satisfies an ordinary

nontrivial generalized polynomial identity (without * ) and hence so does 7?.

By Martindale's theorem, RC is a primitive ring with nonzero socle. By Ka-

plansky's theorem, SV(V) ç RC ç Ly(V) for a self-dual vector space V over

a division ring D endowed with involution. By Proposition 2, the two-sided

Martindale quotient ring of RC is LV(V). Note that the two-sided Martin-

dale quotient ring LV(V) of RC includes the two-sided Martindale quotient

ring Q of 7? as a subring in a natural way. It suffices to show that / van-

ishes on LV(V). Note that Soc(TvC) = SV(V). By Proposition 3, it suffices to

show that / vanishes on Soc(7?C): Suppose that / vanishes on Soc(7?C).

Let qx,q2, ... e LV(V) and let v E V. In computing vf(qx ,q2, ...) = v ,

we need only finitely many relations of the form uq. = u , q*w — w', where

u,u ,w ,w' e V. Obviously, (u ,w) = (uqt,w) - (u,q*w) = (u,w'). Propo-

sition 3 asserts that there exists qj E Soc(RC) such that uq¡ - u and (q¡)*w =

w . Hence v = vf(qx ,q2, ...) = vf(qx , q2 ,...) = 0 .

If C is finite, then there is a nonzero »-ideal 7 of R such that aJ ç R

for all a E C. Then 7 3 77C. But 77C is a nonzero ideal of RC and

hence must contain the minimal ideal Soc(TîC). So / vanishes on Soc(7?C)

as desired.

Now let C be infinite. If * is of the second kind, then R is special by

Proposition 1. The assertion follows from the well-known corresponding result

for ordinary generalized polynomial identities (without *). So we may assume

that * is of the first kind. We proceed by induction on the height of /. Pick

sufficiently but finitely many a E C and let 7 be a nonzero »-ideal of R such

that aJ C I for these a picked. Replace each indeterminate x in / by ax



for these a picked. Then the resulting »-generalized polynomials vanish on 7 .

Note that these a E C picked are symmetric and hence can be moved out of

operation. By the Vandermonde determinant argument, we can solve these

»-generalized identities for the homogeneous parts of /. So each homogeneous

part of / vanishes on 7. It suffices to show that each homogeneous part of

/ vanishes on Q. Note that the height of each homogeneous part of / is not

larger than that of /. Replacing / by one of its homogeneous parts, we may

assume from the start that / is homogeneous in each indeterminate it involves.

Assume that the height of / is 0. Then since / is homogeneous, / must

be multilinear. It is obvious that / vanishes on 7C. Since 7C, a nonzero

ideal of RC, must contain the minimal ideal Soc(7?C), / also vanishes on

Soc(TvC) as desired. So we assume that the height of / is larger than 0. As

the induction hypothesis, we also assume that the assertion of the proposition

holds for any »-generalized polynomials with less height.

Given any indeterminate x in /, we suppress all indeterminates other

than x and write f(x) for / for simplicity of notation. Now consider

g(x + y) = f(x + y) - f(x) - f(y), where y is a new indeterminate. Since g

is of less height than f, g must vanish on Q by our induction hypothesis. So

f(x + y) — f(x) + f(y). We have thus shown that / is additive on Q with

respect to each indeterminate it involves.

Now write / = f(xx, ... ,xn), where xx, ... ,xn are all the indeterminates

/ involves. Set x( = ¿Zjrfotf, where rj0 G 7 and af E C. Using the

additivity of / on Q, we compute

a*,,*,.^»/Íew.-.ew)

7l .In

= £ (^)'"---(c^)h"f(r{X),...,^),

71 .jn

where /?, = the x,-degreeof / (/'= 1,...,«). Since rf e I, f{r^, ... ,rfj)

= 0 and hence f(xx , ... ,xn) = 0. But x( = J~j.r^oA1' are typical elements of

IC. So f vanishes on 7C. Since 7C, a nonzero ideal of RC, must contain

the minimal ideal Soc(7?C), / vanishes on Soc(7?C) as desired.

One might be wondering whether the two-sided Martindale quotient rings

respectively of R and of 7?C are equal or not. If R satisfies a nontrivial

polynomial identity (over C), then 7?^, Q, the left Martindale quotient ring

of RC, and the two-sided Martindale quotient ring of RC are all equal to

RC . But, as the following example shows, this is actually false in general, even

if 7? satisfies a nontrivial generalized identity.


Example. Let D be a (commutative) principal ideal domain and let C be its

quotient field. Suppose that CV is an infinite-dimensional vector space over

C . Fix a basis ¿& = {vx,v2, ...} of CV. For v - J2¡ aiv¡ » u — J2¡ ßtvi G v >

where vi E 3§ and where a¡, ßt E C vanish for all but finitely many i, we

define (v ,u) = J2¡a¡fi¡. Then ( , ) is a nondegenerate bilinear form on VxV.

As usual, let A(V) = End(rF). For a G A(V) such that v.a = £.■<*,,«,,t- l J 'J J

where i>(, VjESS, ai. G C, the adjoint of a, if it exists, is defined by

w a* = IZ/a/,ï,(- Hence, LV(V) consists of all a G /1(F) such that for each

u ■ G «^ , (v(a, Vj) = 0 for all but finitely many i, and SV(V) consists of all

a E A(V) such that v¡a - 0 for all but finitely many i. Let D^ consist of all

aEA(V) suchthat (via,vj)ED for all v^VjESS . Define R = Sv(V)nDoo.

Then RC = SV(V). Also, ideals of R assume the form dR for some d E D.

Hence the left Martindale quotient ring RF of R and the two-sided Martindale

quotient ring Q of R are equal to {a/d: a E D^ and 0 / d E D) and

RF r\Lv(V), respectively. So, if D is chosen so that D ^ C, then A(V) ^ RF

and LK(F)^ß-

IV. Proof of Main Theorem

We begin with the following.

Lemma 8. Suppose that R satisfies a nontrivial »-differential identity. Then R

satisfies a nontrivial generalized identity (without *).

Proof. By linearization, we may assume that R satisfies a nontrivial multilin-

ear »-differential identity 0(x|' ,(xAf) = 0. Assume toward a contradiction

that R does not satisfy any nontrivial generalized identities (without *). By

Theorem 9 [8, p. 77], R does not satisfy any nontrivial »-generalized identities

(without derivations) either. In particular, any linear »-generalized identities

of 7? are trivial. Hence Lemma 4 applies and gives that cp(z¡.,yj.) — 0 is a

multilinear generalized identity of R . Since 0(xf;, (x;A;)*) = 0 is not trivial,

0(z. ,y..) = 0 cannot be trivial either. This is absurd.

If every »-differential identity of R is trivial, then our Main Theorem holds

trivially and there is nothing to prove. So from now on, we assume that 7?

satisfies a nontrivial »-differential identity. By Lemma 8 above, 7? satisfies a

nontrivial generalized identity.

We recall the following [4, p. 68]:

Fact 4. Assume that R satisfies a nontrivial generalized identity. Let p E

D + Dm% be such that p(C) = 0. Then p E 7>nt.

If C is finite, then p(C) = 0 for any p E D + DmX. Hence by the identity

(3), any »-differential identity can be brought to a »-generalized identity without

derivations. So our Main Theorem holds trivially in this case. Thus we may

assume henceforth that C is infinite. By Proposition 1 and Lemma 1, we can

further assume that * is of the first kind.

We recall the following definition [4, p. 64, fourth line from the bottom]: For

px, ... ,pnE D + Dinl, we say that px, ... ,pn are strongly independent if, for

any a; G C, J2p¡cxi E 7>m implies a, — a, = • • • = an = 0.

Lemma 9. Suppose that px, ... ,pn E D + DmX are strongly independent and

that ax, ... ,an E RF . If for all a E C, apl a, +-h afinan = 0, then ax =

••• = a„ = 0. '

Proof. Suppose not. Without loss of generality, we may assume that a, ^ 0.

Choose a basis vx= ax, v2, ... for the subspace ¿~2"=x atC. Express a( (i =

I, ... ,n) in terms of the basis vx,v2, ... : a, = vx and aj = ßtvx + ■ ■ ■ for

/ > 2 . Substituting these expressions into ap'ax-\-Yap"an = 0 and collecting

terms according to vx ,v2, ... , we obtain

(a"' + aßlß2 + aßiß3 + ■ ■ -)vx + (-)v2 + ■ ■ ■ = 0.

Since vx ,v2, ... are C-independent, aMl + aßlß1 + aPlß3 + ■ ■ ■ = 0 follows.

Hence the derivation px + p-,ß2 + p3ß3 + ■ ■ ■ vanishes on C and, by Fact 4,

must be in 7>nt. This contradicts the strong independence of px,p2, ... ,pn .

Lemma 10. If cp is a linear »-differential identity, then the Main Theorem holds.

Proof, cp may be written in the form

77,

, ,, A, . A,,*, V^V^7 (k) A, .(A:) , (k), A,,»,(A'h0 = 0(x ,(x ) ) = 2^£(ß,■ x b,- +c, (x ) d\ )'

I k=\

where A; are distinct regular words in M. Assume that A, is the greatest

among A(.. We prove by induction on A, that if cp(x ', (x ')*) vanishes on a

nonzero ideal of R, then cp(y¡,y*) = 0 is a »-generalized identity on R , where

y are new distinct indeterminates.

If A, =0, then 0 is simply a »-generalized polynomial (without deriva-

tions). The result follows from Proposition 4. So let us assume that Ax ^ <Z.

As the induction hypothesis, we also assume that the assertion of this lemma

holds for any linear »-differential identity whose leading word is strictly less

than A, .

Write A, = àfô(2i] ■ ■ ■ and assume that â\i] = o{2] = ■■■ = ô{s'] ¿ ô^ ,

0 < si < p (p = ch R). We also set Ä, = Ô{2]S{3] ■■■ . Let ß EC. Using the

formula ( 1 ) (in the proof of Lemma 3) and the fact that a* = a for any a E C,

we have

af)(ßx)A'bf) = af'ßx^b^+s^ß^x^bf' + ■■■

and

cf \(ßxf')*d\k) = cf]ß(xArdt]+s,c< V'^Vtff' + ■ ■ ■ -License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Hence

<Kßx) = ß £ 5>í* >A(* >+c\k\xAr df])i k=\

V^V^ n&\ t (le) A,,(fc) (k), Aíx* ,(7C),+ ££57^ ' («, x 'b\ ' + c) \x ') d]') + ■■■

i k = \

oj.1 \ , V^V^ os\'] i (k) ^iu(k) , (*■')/ Ä,.» Ak). ,= ß<t>(x) + 2^l^siß (ai x b, +c)'ix ) d) ') + ■■■•

7 fe=l

Let 7 be a nonzero ideal of 7v on which 0 vanishes. Pick a nonzero ideal 7

of 7? such that 7 ç 7 and ßJ Q I. Then epißx) - ßcpix) vanishes on 7.

Note that the leading word A, of 0(/3x) - /30(x) is strictly smaller than A, .

So the induction hypothesis applies.

We want to collect the terms in epißx) - ßcpix) which contain the regular

derivation word A, : Let px, ... ,pn be all derivations in M other than S\

such that pxAx, ... ,pnAx occur in 0. Let us say pxA, = Ar , ... ,pnAx - Ar .

Then the sum of those terms in 0(/3x) - /30(x) containing A, is of the form

sy"f2(a[k)xAlb\k) + c\k)ixArd[k))k=\

+£rf£(<)^,^)+<',(^r<,)VT=l \k=\ J

By the induction hypothesis,

s^^yb^ + c^y*^)k=\

+ £^Í£(<)<", + <V<))Uor=l \A: = 1 /

is a »-generalized identity for 7?. Since this holds for any ß E C and since

ô\], px, ... ,pn are strongly independent, by Lemma 9,

77|

El (*■') L.(k) , (k) * ,(7Ck „(a, ybx +c\ 'y d\') = 0

k=\

is a »-generalized identity for R . Hence

71,

i V^Y^/ (*) A,,(A') , (Ar), A,,* ,(/,').^ = ££K- x bi +c) (* ) di )7>2 7C=1

also vanishes on 7 . The leading word of 0, is obviously smaller than that of

0. So by the induction hypothesis again,

££(^Vf) + cfV;^)) = oi>2 k = \

is also a »-generalized identity for R. Combining this with the identity

¿(ÍVÍ' + ÍVÍVo,7C=1

the desired result follows.

Using a proof similar to that of Lemma 4, we have

Lemma 11. 7/0 isa multilinear »-differential identity, then the Main Theorem

holds.

Using Lemma 11 and arguing as in the proof of Lemma 6, we have the

following

Lemma 12. If chR — 0, then the Main Theorem holds.

By Lemma 11, we may further assume henceforth that ch R = p > 0, where

p is a fixed prime.

Since 7? satisfies a nontrivial generalized identity, by Martindale's theorem

[7], the socle of RC is nonzero. Set a = Soc(7<C). By the Kaplansky theo-

rem, there exists a self-dual space DV over a division ring D endowed with

involution such that

SviV) E RC ç LviV)

and such that o — SviV). Note that VRC is an irreducible right 7?C-module

and D = Hom(l/RC, VRC). Since Soc(7?C) ^ 0, all the irreducible right 7?C-

modules are isomorphic. The division ring D is also unique up to isomorphism

and is thus called the skew field of RC . Note that, by [7], D is finite dimen-

sional over C.

We need the following from [4, p. 74]:

Fact 5. Let 7 be a nonzero ideal of R . Then ICP 2 o .

Note that the coefficients of »-differential polynomials of R are allowed to

lie in RF . Since the left Martindale quotient ring of RC includes RF as a

subring in a natural way and since, by Proposition 2, both the left Martindale

quotient ring of RC and the left Martindale quotient ring of Soc(TvC) are

equal to A(V), any »-differential polynomial of R can be naturally regarded

as a »-differential polynomial of Soc(7?C). The following lemma is stated in

this sense.

Lemma 13. 7/0 isa ^-differential identity of R, then cp is also a ^-differential

identity of a.

Proof. We prove by induction on the height of 0 that if cp vanishes on a

nonzero *-ideal I of R, then cp vanishes on JCP for some nonzero ideal J of

R . Our desired result will follow immediately from Fact 5.

As in the proof of Proposition 4, pick sufficiently but finitely many a E C .

Let 7 be a nonzero »-ideal of R such that aJ El for those a picked. Replace

each indeterminate x in 0 by apx for those a E C picked. Note that the

resulting »-differential polynomials after such replacement vanish on 7 . Note

that ap are constants of derivations and hence can be moved out of derivations.

Using the Vandermonde determinant argument, we can solve these »-differential

identities of 7 for the homogeneous parts of 0. So each homogeneous part

of 0 vanishes on 7 . Note that the height of each homogeneous part of 0 is

less than or equal to that of 0. Replacing 0 by one of its homogeneous parts

(and 7 by 7 ), we may assume from the start that 0 is homogeneous in each

indeterminate it involves.

If the height of 0 is 0, then 0 must be multilinear. If 0 vanishes on 7, then

obviously 0 also vanishes on ICP as desired. So we assume that the height

of 0 is larger than 0. As our induction hypothesis, we also assume that the

assertion holds for any »-differential polynomial of less height.

Let x be an indeterminate occurring in 0. Suppressing indeterminates other

than x, write 0 = 0(x) for simplicity of notation. Let y be a new indetermi-

nate not occurring in 0. The »-differential polynomial 0(x +y) - 0(x) - cpiy)

also vanishes on 7. Since the height of 0(x + y) - 0(x) - 0(y) is strictly less

than that of 0, by our induction hypothesis, there is a nonzero ideal 7 of R

such that 0(x + y) - 0(x) - cpiy) vanishes on JCP . We have thus shown that

0 is additive on JCP with respect to x .

Let xx, ... ,xn be all indeterminates occurring in 0. Write

0 = 0(X,, ... ,xn).

By the paragraph above, for each i (/' = 1, ... ,n), there exists a nonzero ideal

7( such that 0 is additive on JtCp with respect to x;. Set K = 7 n (f]"=i J,) ■

Assume that the x(-degree of 0 is h¡. Set x( = X^, rj aJ where r. G K and

a- E Cp . Using the additivity and the homogeneity of 0, we compute

*E']'X"EW.Y.AA

=h'iZ(A!)AA:f--A:,ArA':,rj:,...,r_)n >

J.)n

= 0.

Hence cp vanishes on KCP as desired.

By the Jacobson-Noether theorem, the skew field D of R possesses a maxi-

mal subfield O which is separable over C . <X> is hence a primitive extension

of C. Assume that <P = Civ), where v is a primitive element of O over C

satisfying the minimal polynomial

k/(*) = £<***

By the separability of O over C , f (v) = Y.k kakvk~x / 0 .

Any derivation p on C can be uniquely extended to a)" = r" 0 a + r ® a'',

where r G a and a G O. The involution * on a can also be extended to

cr ®c in a natural way, by means of the above extensions

of derivations and involution on o to a®c<&, any »-differential polynomial of

a can be naturally interpreted as a »-differential polynomial of a ®c <X>. The

following lemma is stated in this sense.

Lemma 14. If cp is a ^-differential identity on o, then cp is also a »-differential

identity on o ®c$>.

Proof. Let 0 = 0(x( ' ,(xj ')*) — 0 be a »-differential identity for a. Using

identity (9)' to bring * inside of derivations, we have 0 = 0(x(. ', (x*) ' ) — 0.

Suppressing derivation words for simplicity of notation, we write 0 = 0(x;, x* ).

Set xt = X),a,,'7,., where a; are fixed elements of a and rr are indetermi-

nates intended to range over C. Since * is of the first kind, x* = YL¡ annij ■

Substitute these x¡ and x* into 0 and write the resulting expression in the

following form:

^ £ aiñu • £ a*flu = £ bhlrfij ).

where b are C-linearly independent elements in the left Martindale quotient

ring of o and where cph(r]A) are differential polynomials in ir with coeffi-

cients also in C . If 17. are assigned values from C, then, since J2aijr¡n e °

and (£a/7i7;7)* =£«*-fyeff, 0(Eya,7V Eyflyfy) vanishes. Since ¿> are

C-linearly independent, cp,(r]A.) vanishes on C for each ¿>. Thus cph(t]A) = 0

is a differential identity (without *) for C. By the Kharchenko theorem,

cph(r]rk) = 0 is an identity on C , where rjjjk are new distinct indeterminates.

Since C is an infinite field, (ph{,lljk) is a trivial polynomial. Thus cph(t]A) also

vanishes trivially when r]tj are assigned values in O. So cp(Y^jajjr]jj, £y-a*y-i/,-7-)

vanishes when 17. range over <&. Since a- G o, »7.. G O are arbitrary and

since J2a, 1i ' wnere a; G er, 17. G O are typical elements of o ®c O, 0

vanishes on cr ®c. <P as desired.

As usual, each element in o ®c O can be interpreted as a linear map in

End^F) by defining

v(r ®a) = avr,

where v E V, r Eo , and a E O. Such a representation is faithful by Theorem

2.2 [7, p. 504]. Obviously, V is an irreducible o ®c in End((J)F) is <P itself.

D is finite dimensional over C and hence must also be finite dimensional over

<P. For r E a, the range of r is finite dimensional over D and hence is also

finite dimensional over <P. So each element in a ®c <P is of finite rank in

End(0F). Thus o ®c <P coincides with its own socle.

The bilinear form ( , ) defined on D V is no longer a bilinear form on ^ V,

since the values of ( , ) fall in D but not in <P. However, by the Kaplansky

theorem, there exists a self-dual space U over a division ring A endowed with

involution such that

Su(U)Ca®cOCLu(U).

Since the socle of o ®c <P is nonzero, all the irreducible o ®c <P modules

are isomorphic and hence so are their commuting division rings. Thus we

may assume A = O. Since the two-sided Martindale quotient ring of a ®c <P

includes 7? as a subring naturally, by Proposition 4, it suffices to prove our Main

Theorem for o ®c <P. Replacing 7? by o ®c <P, we may assume henceforth

that the skew field of er coincides with its own extended centroid C.

Now are ready to give

Proof of the Main Theorem. It is well known that any ô E D + 7)int can be

uniquely extended to a derivation on the ring RF of left quotients. See for

example, Exercise 10 [6, p. 101]. The derivation thus extended will also be

denoted by ô. So we assume that any ô E D + D t is defined on the whole

RF.

Suppose that e is an idempotent in o. Then Ve is a finite-dimensional

subspace over C. Let us say that Ve is of dimension m over C. Choose

a basis {vx, ... ,v)n) for Ve over C. Also choose a basis {v( : a > m} for

V(l - e). Then {va: a > 1} is a basis of the whole space V . Let e„ E RF

(= A(V)) be the linear transformation such that v e „ — v„ and v e „ — 0v v y ' a aß p y (\¡i

for y / a . If a, ß < m , then et „ G eae . Also {e„ : a, ß < m} forms a basis

of eae over the field C.

For each S E D + 7>m , we define a new derivation ô on 7^.. (= A(V)) as

follows: Let a E RF be such that vna = J2ßcaßvß- Define ad E A(V) by

setting

vMs) = Hcißvß-

ß

Observe that e„ = 0. Also ô - ó acts trivially on C and hence must be

an inner derivation in 7?^.. Let r& E RF be such that ô = ô + ad(rs), where

ad(rs) is the inner derivation defined by rsE RF . Let M = {S: 8 G M) . For

ö", , ô2 G M, we define »5, < ô-, if and only if ôx < ô2. For A = oxô2-ôn,

where o¡ E M, define A = ¿),<57 • ■ • 8n. Note that if A is a regular word in M,

then A is also a regular word in M.

Suppose that 0 is a given »-differential identity of R. As in the proof of

the generalized Kharchenko theorem (see also the remark following Lemma

4), we may assume that 0 involves only one indeterminate x. Write 0 =

0(x ; ,(x ')*), where A are regular words in M. We may assume that A, >

A2 > • ■ • . For simplicity, let us also assume that {A.: j = 1,2, ...} is closed

under subwords.

Our strategy is to replace each ô E M in 0 by the more concrete derivation

â+ad(rs). But the difficulty is that * is defined only on Q and «oí on the whole

RF . The resulting expression after such substitution may involve (x )* and

r*. Even if x is assumed to range over Q, x may fall out of Q. Similarly,

rs may possibly not be in Q. In either case, the expressions (x )* and r*ö

make no sense.

Our solution is as follows: Using the identity (9), we bring 0(x ' ,(x ')*) to

0(x ' , (x*) '), where A* are regular words in M*. Observe the following: For

aEC and for ô E D + £>im ,

(S~) ,, *xc5n* / ¿,* àöl =((a ) ) = (a ) = a ,

since * is assumed to be of the first kind. Hence ô* - ô vanishes on C and

so does S* - 5 . Thus there exists ts E Rf such that ö* - ô + ad(ts).

Now, in cp(xAj, (x*)Aj ), we substitute ô + ad(rg) for ô in A, and ¿-r-aa7^)

for S* in A* respectively. Using identities (l)-(6), the resulting expres-

sion of such substitution can be brought to a new »-differential polynomial

y/(x ', (x*) '), where A. are regular words in M.

Define A, (/ > 1) inductively as follows: For 7 = 1, define A, = the highest

total degree of xA' and (xA|)* in 0. For j > 1 , define A, = the highest total

degree of x ; and (x J)* in those monomials of 0 whose total degree of x '

and (x ')* is A( for each i < j.

We define the leading part 0o(xA; ,(xAj)*) of 0(xAj ,(xA;)*) to be the sum

of those monomials of 0 whose total degree of x ' and (x ')* is A for each

j > 1. Similarly, we define the leading part y/0(x ' ,(x*) ') of y/(x ' ,(x*) ')

to be the sum of those monomials of y/ whose total degree of x ' and (x*) '

is A for each j > 1. It is important that the generalized polynomial 0o(zj. ,yß

coincides with the generalized polynomial y/0(z.,y ), where z., y. are new

indeterminates.

Set x — y.^. „s e «Ç „, where C tí are indeterminates intended to range¿—M <a ,/)<777 aß^ap ' ^aß °

over C and hence are assumed to commute with 7?,-. Since * is of the first

kind, we have x* = Ei<a>fi<m^C^ ■_ Suppose that A^= ô{ô2-■ ôsôs+x-■■

is a regular word in M, where Sx - ô2 = • ■ • = ös ^ ôs+x and 0 < s < p

(p = ch R). Since ea „ — 0 for ô E M, we have

A _ V—* „A

X ~~ /-*. eaß âß ■

\<a,ß<m

Using the formula ( 1 ) in the proof of Lemma 3,

Ä

(x*)A = £ e*«ßt«ß

\\<a,ß<m

= £ (<ß£ß+s(<ßfCf- + ••■),l<a,/j<í?7

where the dots above denote a sum of terms (e*„) (Ç „) > where A , A <

<52<53 • • • . Substitute these expressions for x and (x*) into y/(x ', (x*) ')

and let t](Ç\) be the »-differential polynomial thus obtained.

We define the leading part »/0(í„«) to he the sum of those monomials of 77

whose total degree of Cri« (I < a,ß < m) is h ■ for each j > 1. Let Ç( „. be

new indeterminates. Observe that

%^t*ßj> ~ ^0 I ¿^ enß^nßj> Z-7 eaß*aßj\\<a,ß<m \<a,ß<m

— <rr)\ 2-7 eaßl*aßj> Z_j eaßâßj

\j<rt,/l<777 \<a,ß<m

Now write r¡(C,L) = J2V vrl,Xâß) > where v E RF are C-linearly independent

and where r]v(Çni) are differential polynomials in Ç „ with coefficients in C.

When £„ are assigned values from C, r](Çi) assumes the constant value

0. Since v are assumed to be linearly C-independent, f],,(ClA) = 0 for each

v. Thus, for each v, %,(C„Jß) = 0 is a differential identity on C. By the

Kharchenko theorem, »/,,(£„«,■) = 0 is a generalized polynomial identity on C ,

where C„», are new indeterminates. Hence f/(£ „.) = 0 is also a generalized

identity on C (with coefficients in RF ). Since C is assumed to be infinite,

the leading part 10(Cltßß = 0 of r}(C,iißß = 0 is also a generalized identity on

C. Since

fu(£«/?./) = 00 I ¿^ eaßâßj' Z_^ eaßâßj\l<<!,/><777 I<«,/j<777

and since e„ (1 <a,ß<m) span eae over C , we obtain that cpQ(y ,y*) = 0

for yj e eoe .

Given any finitely many yx,y2, ... E o, there exists an idempotent e E a

such that yx,y2, ... G eoe . By the result of the previous paragraph, cp (y. ,y*)

- 0. Thus we have shown that 0o(y¡.y*) = 0 is a »-generalized identity on

a . By Proposition 4, 0O(;P ,y*) = 0 is also a »-generalized identity on Q .

Now consider

cp\xA', ixA'f) = 0(*ÂJ, (*V) - U*Aj - (*AT) •

Let 0q(x ', (x J)*) be the leading part of 0'(x ', (x ')*) • Repeating the above

argument, we can show similarly that cp'0iyj,y*) vanishes on Q. Continu-

ing in this manner, we can prove finally that the Main Theorem holds for

0(x\(xAO*).

Remark 1 . In our Main Theorem above, derivations are assumed to be in

Ant + C Der7?. However, as remarked in [4, p. 74], DerR may be replaced

by the larger set consisting of all derivations p E Der RF for which there exists

a nonzero ideal I of R such that IM ç R. Also the given differential identity

can be assumed to vanish only on a nonzero ideal of 7? instead of on the whole

ring R . The assertion of the Main Theorem above (together with its proof) still

holds for such a generalization.

Remark 2. As in [3] and [4], we have deliberately avoided the following two

fundamental questions: ( 1 ) With respect to a basis M of D linearly ordered by

<, does a differential polynomial give rise, via the basic identities (l)-(9), to a

unique differential polynomial of the form 0(x( ;, (*,J)*) > where A are distinct

regular derivation words in M (with respect to < ) and where 0(z(.., z*) is a

»-generalized polynomial? (2) Is the nontriviality of a »-differential polynomial

independent of the choice of M and the linear order < on it? Both questions

have affirmative answers. Actually, we can give a free-product treatment of *-

differential polynomials as initiated in [5], where all the basic notions can be

made precise and then the two questions above can be proved mathematically.

As such a treatment is rather long and does not seem to be immediately relevant

to our main theme here, we take up these matters somewhere else.

References

1. I. N. Herstein, Rings with involutions. Univ. of Chicago Press, Chicago, 1976.

2. N. Jacobson, Structure of rings. Amer. Math. Soc, Providence, R.I., 1964.

3. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika 17 (1978), 220-238.

4. _, Differntial identities of semipnme rings. Algebra i Logika 18 (1979), 86-119.

5. C. Lanski, Differential identities in prune rings with involution, Trans. Amer. Math. Soc. 291

(1985), 765-787.

6. J. Lambek, Lectures in rings and modules, Chelsea, 1976.


7. W. S. Martindale III, Prime rings with involution and generalized polynomial identities, J.

Algebra 22 (1972), 502-516.

8. L. H. Rowen, Generalized polynomial identities, J. Algebra 34 (1975), 458-480.

9. _, Polynomial identities in ring theory, Academic Press, New York, 1980.

Department of Mathematics, National Taiwan University, Taipei, Taiwan 10764,

Republic of China


♦ DIFFERENTIAL IDENTITIES OF PRIME RINGS WITH ......Key words and phrases. Differential identity,...

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