PACIFIC JOURNAL OF MATHEMATICSVol. 95, No. 1, 1981

ON RINGS WITH BOUNDED ANNIHILATORS

JOHN A. BE ACHY AND WILLIAM D. BLAIR

Commutative rings for which the annihilator of eachfinitely generated module is determined by n or fewer ele-ments are studied. For Artinian rings the bound n is ex-plicitly determined. Noetherian rings satisfying this condi-tion are of Krull dimension one. A new characterizationof Dedekind domains is given, and the relationship betweenthe bound and the number of generators of ideals is studied.

It is well-known that for commutative Noetherian rings there isa one-to-one correspondence between prime ideals and isomorphismclasses of indecomposable, injective modules. Gabriel [6] showedthat this correspondence still holds for any noncommutative Noe-therian ring R which satisfies the following condition (H): for eachfinitely generated module BM there exist elements ml9 m2, , mn e Msuch that AnnΛ(Λf) = Ann^m^ m2, , m j . (Here AnnΛ(S) = {r eR I rs = 0 for all seS}, for any subset S £ M.) Cauchon [4] showedthat the converse holds, and thus condition (H) holds, for example,for any noncommutative Noetherian ring which satisfies a polynomialidentity or is integral over its center.

If the module RM has the descending chain condition on annihi-lators Ann(S) of subsets S Q M, then it is clear that AnnΛ(Λf) =AnnΛ(m!, m2, , mn) for elements ml9 m2, , mn e M. Thus the classof left Artinian rings is one of the basic classes for which GabrieΓscondition (H) holds. In fact, the ring R is left Artinian if and onlyif the condition on annihilators holds for all left iϋ-modules, not justthe finitely generated modules (see [2]). If R is left Artinian, wenote that the number of elements required in condition (H) is bound-ed by the length of a composition series for the ring R. This mo-tivates the following definition.

DEFINITION. An iϋ-module M is said to have bound n if n isthe least integer such that AnnΛ(Λf) = AnnΛ{mlf -—,mn} for somefinite subset {ml9 , mJ £ M. A ring R is said to have (global)bound n, denoted by bd(R) = n, iί n is the supremum of the boundsof all finitely generated i2-modules.

We note that an i?-module M has bound n if and only if n isthe least integer such that there exists an exact sequence 0 —>R/AnnR(M) —> Mn, where Mn denotes the direct sum of n copies ofM. This fact will be used repeatedly in the paper.

2 JOHN A. BEACHY AND WILLIAM D. BLAIR

We have been able to compute the bound of a commutativeArtinian ring in terms of the lengths of homogeneous componentsof the socles of factor rings, but a similar characterization for non-commutative Artinian rings appears to be much more difficult toobtain. In this paper we restrict our study to commutative Noe-therian rings, in which case the bound of a ring can be related tothe number of generators required for ideals of the ring. We note,however, that a ring with a finite bound need not be Noetherian. Infact, if R is commutative and the module RM is a direct sum(BaeIRma of cyclic submodules Rma, then AnnΛ(Λf) = AnnΛ(m) forthe element m = (ma)aeI e M. Thus any commutative ring whosefinitely generated modules are direct sums of cyclic modules hasbound 1, and such rings need not be Noetherian (see Brandal [3]).

We show that a commutative Noetherian ring with a finitebound must have Krull dimension 1. In this case, the ring hasbound 1 if and only if it is a ZPI-ring, and so a Noetherian domainis a Dedekind domain if and only if it has bound 1. For a localNoetherian domain, the existence of a finite bound is equivalent tothe existence of a bound on the number of generators required forany ideal of the ring, and in this case the two numbers are equal.For a nonlocal domain, the two numbers differ by at most one. Inthe last part of the paper we consider the problem for rings otherthan domains by using a weaker notion of bound.

All rings under consideration will be assumed to be commutativerings with unity, and all modules will be assumed to be unital. Theinjective hull of a module RM in the category of iϋ-modules will bedenoted by ER(M).

We will make extensive use of the notion of the Goldie dimen-sion of a module. (The reader is referred to Goldie [7] for details.)A nonzero module is said to be uniform if any two nonzero sub-modules have a nonzero intersection. A submodule N of M is es-sential in M if N nontrivially intersects any nonzero submodule ofM. A module M has Goldie (or uniform) dimension n if M containsa direct sum of n nonzero submodules, but no longer direct sum.Then M has Goldie dimension n if and only if M contains an es-sential direct sum of n uniform submodules, or equivalently, if andonly if E(M) is a direct sum of n indecomposable submodules. Itis evident that the Goldie dimension of a submodule of M is lessthan or equal to that of M.

We begin with an easy but useful lemma.

LEMMA 1. ( i ) Let M be an R-module. If the Goldie dimen-sion of R is n and 0 —> R —> Mι is an exact sequence, then there isa homornorphism such that 0 —> R —> Mn is exact.

ON RINGS WITH BOUNDED ANNIHILATORS 3

(ii) If bd(R) = n, then bd(R/I) ̂ n for each ideal I of R.

(iii) bd{Rλ © - © Λ.) - max^,*. {fecZCR,)}.

(iv) Lei S be a multiplicative set. Then bd(Rs) ^ bd{R).

Proof. ( i ): Let (m^ , mt) be the image of 1 under R -> Λf*,and then note that {/,,}, j" = 1, , t is a direct family where I3 =Πi^i AnnΛ{mJ.

(ii) and (iii) are clear.(iv): Let M be an ^-module generated by ^, , xt. Consider-

ing M as an iί-module, let N be the jβ-submodule generated byxl9 , xt. Then there exist ml9 , mM 6 iV such that Ann^m^ ,mn) = Ann^AΓ), where τ& = bd(R). But Ann^m^ , m j = AτmBs(M)and so bd{Rs) ^ w.

Recall that the socle, Soc(Λf), of a module M is the sum of itsminimal (=£0) submodules. The socle is a direct sum of certain ofthe minimal submodules. A homogeneous component of the socle isthe sum of all minimal submodules in an isomorphism class of mini-mal submodules.

THEOREM 2. Let R be an Artinian ring. The bound of R isthe supremum of the lengths of the homogeneous components of thesocle of R/A, for all ideals A of R.

Proof, Observe that the supremum exists since the length ofR/A is at most the length of R for all ideals A of R.

Let n be the supremum of the lengths of the homogeneous com-ponents of the socles of all factor rings of R. We first show thatbd(R) ^ n. Suppose that R = R/A has a homogeneous component ofits socle of length n, and let / be a minimal ideal of R which ap-pears as a direct summand of the socle S of R exactly n times. LetX be a direct sum of one minimal ideal from each isomorphismclass of minimal ideals of R. We consider the injective hull E =ER(X) and show that 0 —• jβ —> En is exact and that n is the leastnumber of copies of E into which R can be embedded.

Since each homogeneous component of S has at most n sum-mands, there exists an exact sequence 0 —> $ -> En. Since E is in-jective and since S is essential in R, this extends to an exact sequ-ence 0-+E(R)-^En, and hence R embeds into En. If 0-^R->Em

is exact for some m < n, we have that Em contains a direct sum ofn copies of / and thus Xm = Soc(Zm) = Soc(£rm) would contain In, acontradiction to the Krull-Schmidt theorem.

Let (a?!, •••,»») be the image of 1̂ in En, and let M be the Rsubmodule of E generated by {xlf ••-,&»}. Then M is a faithful,

4 JOHN A. BEACHY AND WILLIAM D. BLAIR

finitely generated i?-module such that R embeds into n but no fewercopies of M. Hence bd(R) Ξ> bd(R) ^ n. (We remark that it is notnecessary to pass to M since X is a finite sum of minimal ideals, Eis a finite direct sum of indecomposable, injective .β-modules andhence is itself finitely generated by a theorem of Matlis.)

To show that bd{R) ^ n, we note that R = R1 φ 0 Rt is adirect sum of local Artinian rings and since bd{R) = m.SLx{bd(Rt)}9 itsuffices to show that bd(Rt) ^ n where n is defined as above. Sinceeach homogeneous component of Soc(i2/A) is contained in some Rt/ARi9

it suffices to show that bd(R) ^ n when R is a local Artinian ring.Let M be a finitely generated jR-module and R = R/AnnR(M).

Since R is local, all minimal ideals are isomorphic and the socle ofR is homogeneous. Since every nonzero ideal contains a minimalideal, Soc(R) is an essential direct sum of isomorphic minimal ideals,and thus the Goldie dimension of R is equal to the length of SocCB).Since we have an embedding 0 —> R —> Mt where t is the length ofR, there exists an exact sequence 0 —> R —> Mk where k is the lengthof SOC(JB) by Lemma 1. Since k ^ n, there is an exact sequence0 -> R -> Mn and bd{R) ̂ n.

The following corollary will be useful in our study of Noetherianrings with a finite bound.

COROLLARY 3. Let R be a local Noetherian ring with maximalideal m. // άiΏiB/m(mk/mk+1) = n for some k e Z+, then bd{R) ̂ n.

Proof. Since R/mk+1 is Artinian and since the (homogeneous) lengthof the socle of R/m^1 is a least n, we have bd{R)^n by Theorem 2.

Our next result shows how severe a restriction the existence ofa bound places on the class of Noetherian rings.

THEOREM 4. Let R be a Noetherian ring. If R has finite bound,then the Krull dimension of R is at most one.

Proof. Since the condition that R has a finite bound passes toall localizations, we may assume that R is a local ring. Let m de-note the maximal ideal of R and let bd(R) = n. By Corollary 3,dimΛ/m(m7πt*+1) ^ n for all keZ+. Let f(k) be the length of R/mk.For sufficiently large fe, f(k) is a polynomial in k (Hilbert-Samueltheorem). The degree of f(k) is less than or equal to 1 sincef(k + 1) — f(k) is the length of mk/mk+ί, which is bounded by n. ByTheorem 11.14 of Atiyah-MacDonald [1], we have that the Krulldimension of R is at most 1.

ON RINGS WITH BOUNDED ANNIHILATORS 5

A ring R is called a ZPI-ring (Zerlegung Primideale) if everyideal can be written as a product of prime ideals of R. R is aZPI-ring if and only if R is a Noetherian ring such that for eachmaximal ideal m of R there are no ideals of R strictly between mand m2. (See Larsen and McCarthy [9].)

THEOREM 5. R is a Noetherian ring with bd(R) = 1 if andonly if R is a ZPI-ring.

Proof. Let R be a Noetherian ring with bd{R) = 1 and let mbe a maximal ideal of R. Then bd(Rm) = 1 and Corollary 3 impliesάimRm/mRm(mRJm2Rm) ^ 1. Thus there are no ideals of Rm strictlybetween mRm and m2Rm, and so there are no ideals of R betweenm and m2, since such ideals are m-primary.

Conversely, by Theorem 9.10 of Larsen and McCarthy [9], if Ris a ZPI-ring, then R is a finite direct sum of Dedekind domainsand special primary rings. Since any finitely generated module Mover a Dedekind domain D is of the form M = Ώ\IX φ © Djlt

where Ix >̂ J2 ^ ^ /*, we have that It — Ann^Λf) and thus thereis an embedding of D/AnnD(M) into M. Hence bd(D) = 1. If S isa special primary ring, then the socle of S/I for any ideal / haslength 1 and thus by Theorem 2, bd(S) = 1. Hence bd(R) = 1.

By Theorem 4 of Warfield [10], R is a Noetherian ring withbd(R) = 1 if and only if every finitely generated R module is adirect summand of a direct sum of cyclic j?-modules. We also pointout that a special case of Theorem 5 gives a new characterizationof Dedekind domains.

COROLLARY 6. R is a Dedekind domain if and only if R is aNoetherian domain and bd{R) — 1.

It is well known that every ideal in a Dedekind domain can begenerated by two elements. As an extension of this last corollarywe explore the relationship between Noetherian domains of finitebound and the number of elements required to generate their ideals.I. S. Cohen [5] designates the rank of a ring to be n if every idealof R can be generated by n or fewer elements. According to thisdefinition a ring of rank k is also of rank h for all h ^ k. We willsay that R has least rank n, denoted by rk(R) — n, if it has rankn but not rank n — 1.

THEOREM 7. Let R be an Artinian ring. Then bd(R) = rk{R).

6 JOHN A. BEACHY AND WILLIAM D. BLAIR

Proof, Suppose that rk(R) = n. Then R = R, 0 0 Rf is adirect sum of local Artinian rings and rk(Ri) ^ n. Let A be anideal of Rt. Then the socle of RJA can be generated by n or fewerelements and it is homogeneous, so the length of the socle is atmost n. Hence bd{Rτ) S n by Theorem 2 and bd(R) ^ n.

Conversely, suppose that bd(R) = n. Then R = Rλ 0 0 Rt isa direct sum of local Artinian rings with maximal ideals tπ, andbd(Ri) ̂ n. Let / be an ideal of Rt which requires k generators.Thus I/mJ is a λ -dimensional vector space over R/m^ Hence thering R/mJ has Goldie dimension at least k, and so the homogeneouslength of the socle of R/mJ is at least k. Hence bd(Rt) ^ k, andso bd{R) Ξ> k. Thus n — bd(R) ^ rk(R), and the theorem is proved.

THEOREM 8. Let R be a Noetherian domain. Then R has finiterank if and only if R has finite bound, and in this case rk{R) =bd{R) or rk{R) = bd(R) + 1. If R is a local Noetherian domain,then rk(R) = bd(R).

Proof. If R has finite rank or finite bound, then the Krulldimension of R is one by Theorem 9 of Cohen [5] and Theorem 4.Thus all proper factor rings of R are Artinian.

Suppose that rk{R) — n. Let M be a finitely generated jβ-module.If Anni2(M) = 0, then since R is a domain we have an exact se-quence 0 —> R -> M. If Ann(M) Φ 0, then R\kxmR{M) is an Artinianring and since rk(R/AnnB(M)) = bd(R/ArmR(M)) ^ n, R/AnnR(M) em-beds into n copies of M. In any case, there is an exact sequence0 -> RIAnnR(M) -> Mn and so bd{R) ̂ rk{R).

Suppose that bd{R) = n. If / is a nonzero ideal of R, thenchoose 0 Φ a 6 I. Then R/aR is Artinian and bd(R/aR) ̂ n. Thusrk(R/aR) :g n and / can be generated by n + 1 (or fewer) elements.Thus rk(R) ^bd(R) + 1.

Now suppose that R is a local Noetherian domain with maximalideal m and bd(R) = n. If / is an ideal of R requiring k generators,then dimΛ/w(7//m) = k. Thus the Goldie dimension of R/Im is atleast k and so the homogeneous length of the socle of R/Im is atleast k. Since R/Im is Artinian, bd(R/Im) ̂ k and thus n ^ k andbd{R) ^

COROLLARY 9. Let R be a local Noetherian domain. Then Rhas finite bound if and only if the Krull dimension of R is atmost one.

PROPOSITION 10. Let R be a Noetherian domain. If bd{Rm) ^ nfor all maximal ideals m of R, then bd(R) ^ n.

ON RINGS WITH BOUNDED ANNIHILATORS 7

Proof. We first observe that if Rm has finite bound for allmaximal ideals m, then R has Krull dimension at most one. Let Mbe a finitely generated iϋ-module, with A = AnnΛ(Λf). If A = 0,then there exists an embedding 0 -^ ϋ! -* ikf. If A ^ 0, then R/A isArtinian and R/A = RJc\1RH 0 0 RJqtRn where A = q, Π Π <\t

is a primary decomposition of A and each q* is ^-primary. Thusthere exists an embedding 0 —> i?/A —• M% since bd(RH) ^ w for eachi. In any case, iϊ/A embeds into n copies of M, and hencebd(R) ^ w.

Recall that if R is a domain with quotient field K, then anysubring S with RQS^K is called an overring of R. The followingtheorem extends the well-known result that any overring of aDedekind domain is a Dedekind domain.

THEOREM 11. Let R he a Noetherian domain with finite bound,and let K he the quotient field of R. If S is an overring of R,then bd(S) ^ bd(R).

Proof. By Proposition 10 it is sufficient to show that bd(Sn) <£bd(R) for all maximal ideals of S. Let m be a maximal ideal of Sand let m' = m n R Then Sm is an overring of Rm, and both ringshave Krull dimension at most one by Theorem 93 of Kaplansky [8].The proof of that theorem shows that if A is a nonzero ideal of Sm9

then SJA can be embedded in a proper cyclic i2m/-module, and sothe Goldie dimension of SJA is at most that of the cyclic i?m/-module.Since Sm and Rm, are local, the socle of any proper factor ring ishomogeneous, and so the length of the socle of SJA is at mostbd{Rmr) ^ hd(R) and hence bd(Sm) ^ bd(R).

Next we wish to consider Noetherian rings which are not neces-sarily domains. For this purpose we introduce the notion of aweak bound.

DEFINITION. The ring R has weak bound at most n, denotedwbd(R) ^ n, if for every finitely generated ϋJ-module M such thatAτmR(M) has no embedded primes there exist n elements mlf ,mn e M such that Ann^M) = Ann^m^ , mn}. If wbd(R) ^ n butwbd(R) < n, then we write wbd(R) = n.

PROPOSITION 12. If R is a Noetherian ring and wbd(R) ^ n,then the Krull dimension of R is at most one.

Proof. We first observe that if S is a multiplicative set in R,

8 JOHN A. BEACHY AND WILLIAM D. BLAIR

then wbd{R) ^ n implies wbd{Rs) ^ n. To see this let M be afinitely generated ϋ^-module whose annihilator has no embeddedprimes. If {xl9 •••, xt) is a set of generators of M, then as in theproof of Lemma 1 let N be the iϋ-submodule of M generated byxlf •••, xt. Then AτmB(N) = AτmRs(M)Πi2 has no embedded primes,and so there exist mlf , mneN such that Ann^iV) = Ann^m^ ,m j . Thus AτmBs(M) = Ann* {m^ , m j .

The proof of the proposition follows from Corollary 3 exactly asin the proof of Theorem 4 because wbd{R) = bd{R) for Artinianrings.

COROLLARY 13. If R is a Noetherian domain, then wbd(R) =bd(R).

LEMMA 14. Let p be a prime ideal of the Noetherian ring R,and suppose that R/p has finite rank. Then R/pk has finite rankfor each ke Z+.

Proof. The proof is by induction on k. The case k = 1 is trueby hypothesis. Assume that R/p1'-1 has finite rank. Thus everyϋJ-submodule of R/p16'1 can be generated by rk{Rjpk~ι) elements. TheRjp-moάxήe pk~ιjpk is generated by say, m elements. Thus any sub-module of pk~ιjpk can be generated by m rk(R/p) elements. Henceany ideal of R/pk can be generated by rk{Rlpk~ι) + m rk(Rjp) ele-ments and thus R/pk has finite rank.

THEOREM 15. Let R be a commutative Noetherian ring. ThenR has finite rank if and only if R has finite weak bound.

Proof. Suppose that rk(R) = n and suppose that R has t mini-mal prime ideals. Let M be a finitely generated i?-module whoseannihilator has no embedded primes. Let AnnR(M) = qt Π ΓΊ qs Πqs+1Π nqfe be a primary decomposition of AnnR(M), where qt is^-primary and plf •••,£, are minimal prime ideals of R while ps+1, , pk

are maximal ideals of R. Since R has Krull dimension at most oneand AnnB(M) has no embedded primes, we have that q1 Π Π qs andcjs+i Π Γl qk are comaximal ideals for if m is a maximal ideal con-taining q4 Π ΓΊ qs and qs+1 Π Π qk9 then m contains some qt with1 ^ i ^ s and some qd with s + 1 <; j ^ k and so m = p3- 2 t>< con-tradicting the fact that ArmR(M) has no embedded primes. Thus

R/ArmR(M) = Rj{qλ ΓΊ Π q.) Θ Λ/(q.+i Π Π qfc) .

Since R/(qs+1 ΓΊ Π qfc) = R/qs+i © © Rfak is Artinian and sincerk(R) = n, we have that the bound and hence the weak bound of

ON RINGS WITH BOUNDED ANNIHILATORS 9

R/(qs+1 Π Π qk) is at most n.On the other hand, since for i = 1, , s, q, — qιRH Π R, we have

that R/qt embeds into RH/qiRH where R94 is the Artinian local ringobtained by localizing R at pt. Since rk(RH) ^ n, the Goldie dimen-sion of any factor ring of RP4 is at most n. Noting that the Goldiedimension of RHjqiRH as a ring is the same as the Goldie dimensionof RHlc\iRH considered as an i2Pi-module, which is the same as theGoldie dimension when it is considered as an ϋ?-module, we see thatthe Goldie dimension of R/qt as an ϋϊ-module is at most n. Sinces 5̂ t and since Rl{qι Π ΓΊ qs) is contained in R/qx φ φ R/qs, wesee that the ring Rj{qx Π Π qs) has Goldie dimension at most t n.Thus i?/(<li Π Π Cfβ) embeds into nt copies of any faithful moduleover it. Therefore R/AτmB(M) embeds into nt copies of M, sinceM = (R/iq.n Π qs))ikf θ (Λ/(q.+i Π Π qk))M and (Λ/(qx Π Π q.))Mis a faithful Rl{qx Π Π qs)-module and (R/(qs+1 ΓΊ Π Q*))-M" is afaithful Λ/(qβ+1 Π Π qA)-module. Thus wbd(R) ^ wί.

Conversely, suppose that wbd(R) = n. Then for each prime idealp of R, R/p is a Noetherian domain and bd(R/p) = wbd(R/p) <; w.Thus rk(R/pk) <: w + 1. By Lemma 14, rk(R/pk) < oo for each A; 6 Z + .

From a primary decomposition of 0 we see that there exists aninteger k such that 0 = Πi=i $ where the pt are the minimal primesof R. Hence we have an exact sequence

0 > R > R/pϊ φ 0 RlPk

t .

Since rk{Rjp\) is finite we see that rk(R) is finite.

THEOREM 16. Let R be a semilocal Noetherian ring of dimen-sion one. Then wbd(R) ^ bd(R) < o°.

Proof. R has only finitely many prime ideals, say k. Let A bean ideal of R and suppose that A = qx n Π qt is a primary decom-position of A, where qt is ^-primary. Thus 0->J?/A-^i2/qxφ ••• φR/qt is exact, and as in the proof of Theorem 15, the Goldie dimen-sion of R/A is less than or equal to k-ma.x{rk(Rfi)}. Each RH hasfinite rank by Corollary 9 and an application of Lemma 14, as inthe last part of the proof of Theorem 15.

Finally we mention that we have no example of a ring forwhich the weak bound is finite but the bound is not finite.

REFERENCES

1. M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA, 1969.

10 JOHN A. BEACHY AND WILLAM D. BLAIR

2. J. A. Beachy, On quasi-Artinian rings, J. London Math. Soc, (2) 3 (1971), 449-452.3. W. Brandal, Commutative rings whose finitely generated modules decompose, LectureNotes in Math., 723 (1979).4. G. Cauchon, Les T-anneaux et la condition de Gabriel, C. R. Acad. Sci. Paris, 277(Serie A) (1973), 1150-1156.5 I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J.,17 (1950), 27-42.

6. P. Gabriel, Des categories abeliennes, Bull. Soc. Math. France, 90 (1962), 323-448.7. A. W. Goldie, The structure of Noetherian rings, Lecture Notes in Math., 246 (1972),214-321.8. I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.9. M. D. Larsen and P. J. McCarthy, Multiplicative Theory of Ideals, Academic Press,New, York, 1971.10. R. B. Warfield, Jr., Decomposability of finitely presented modules, Proc. Amer. Math.Soc, 25 (1970), 167-172.

Received April 18, 1980.

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