Rings and modules with chain conditions up to isomorphism

Rings and modules with chain conditions up to isomorphism

Zahra Nazemian, Murcia University, Spain

Graz, January 2021

Based on joint works with Alberto Facchini

Artinian dimension and isoradical of rings, J. Algebra, 2017.

Modules with chain conditions up to isomorphism, J. Algebra 2016.

On isonoetherian and isoartinian modules, Contemp. Math., Amer. Math. Soc. 2019

One can try to study a similar generalization for other algebraic structures.

For example we did it for commutative monoids:

• Chain conditions on commutative monoids (with B. Davvaz), Semigroup Forum, 100 (2020), 732–742.

Emmy Noether, 1882- 1935

Noetherian ring: A ring is called right noetherian if for every ascending chain of right ideals of , there exits an integer number such that

for every

R

I1 ≤ I2 ≤ ⋯ Rn

In = Ii i ≥ n .

Emmy Noether, 1882- 1935

Noetherian ring: A ring is called right noetherian if for every ascending chain of right ideals of , there exits an integer number such that

for every

R

I1 ≤ I2 ≤ ⋯ Rn

In = Ii i ≥ n .

Noetherian modules: A module is called noetherian if for every ascending chain

of submodules of there exists such that for every

M

M1 ≤ M2 ≤ ⋯ Mn Mn = Mi

i ≥ n .

Emil Artin , 1898 - 1962

Artinian Modules: A module is called artinian if for every descending chain

of submodules , there exists such that for every

M

M1 ≥ M2 ≥ ⋯M n Mn = Mi

i ≥ n .

Every right Artinian ring is right Noetheran.

Emil Artin , 1898 - 1962

Artinian Modules: A module is called artinian if for every descending chain

of submodules , there exists such that for every

M

M1 ≥ M2 ≥ ⋯M n Mn = Mi

i ≥ n .

2016, Facchini and Nazemian, Isoartinian modules:


A module is called isoartinian ifM


A module is called isoartinian ifM for every descending chain

of submodules of ,M1 ≥ M2 ≥ ⋯ M



of submodules of ,M1 ≥ M2 ≥ ⋯ M there exists such thatn

for every Mn ≅ Mi i ≥ n .


Right isoartinian ring:





Right isoartinian ring: If is an isoartinian module, the ring is called right isoartinian.

RR R





Right isoartinian ring: If is an isoartinian module, the ring is called right isoartinian.

RR R

You can define left isoartinian similarly. Also talk about isoartinian ring.




2016, Facchini and Nazemian, isonoetherian modules:

A module is called isonoetherian if for every ascending chain of submodules of , there exists such

that for every

MM1 ≤ M2 ≤ ⋯ M n

Mn ≅ Mi i ≥ n .

2016, Facchini and Nazemian, isonoetherian modules:

A module is called isonoetherian if for every ascending chain of submodules of , there exists such

that for every

MM1 ≤ M2 ≤ ⋯ M n

Mn ≅ Mi i ≥ n .

Right (left) isonoetherian ring:

If ( ) is isonoetherian module, the ring is called right (left) isonoetherian.

RR RR R

Also we can talk about isonoetherian ring.

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Isosimple modules:

Recall that a nonzero module is called simple if it has no nonzero proper submodule.

M

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Isosimple modules:


M

We call a nonzero module isosimple if every nonzero submodule of is isomorphic to .

MM M



Isosimple modules:


M


MM M

Every submodule in an isosimple module is cyclic.



Isosimple modules:


M


MM M

Every submodule in an isosimple module is cyclic.

If is isosimple, where is a domain, then is a PRID. DD D D



Examples:

Consider the abelian group . ℤ



Examples:


All nonzero submodules of are isomorphic to , that is, every submodule of is in form for some integer . We know . So is isoartinian, isosimple and isonoetherian.

ℤ ℤN ℤ mℤ m

mℤ ≅ ℤ ℤ



Examples:


A PRID is isosimple, isonoetherian and isoartinian.



Examples:


Consider any vector space , where is field. Vk k




Examples:



It is always isoartinian even if it is of infinite dimension.






Examples:







Indeed if is a descending chain of submodules, then we

V1 ≥ V2 ≥ ⋯



Examples:





V1 ≥ V2 ≥ ⋯




have a descending chain of cardinals dim(V1) ≥ dim(V2) ≥ ⋯,



Examples:




we know that every descending chain of cardinals stops somewhere, and two vector spaces over are isomorphic if and only if they are of the same dimension.

k





V1 ≥ V2 ≥ ⋯have a descending chain of cardinals dim(V1) ≥ dim(V2) ≥ ⋯,



Examples:



It is isonoetherian if and only if it is of finite length.






Examples:



Free modules over a PID (principal ideal domain).






Examples:



If is an infinite dimensional vector space, the ring is a

right isoartinian ring which is not right isonoetherian.

Vk (k Vk

0 k )









Examples:

In the following, I will give you more examples….









If is an infinite dimensional vector space, the ring is a

right isoartinian ring which is not right isonoetherian.

Vk (k Vk

0 k )



Artinian modules are essential extensions of their socle.




Every artinian module contains a simple submodule.





So, every artinian module contains an essential submodule which is a finite direct sum of simple modules.






Recall that is called essential in if 0 ≠ N ≤ M M







for every nonzero N ∩ N′ ≠ 0 N′ ≤ M .







for every nonzero N ∩ N′ ≠ 0 N′ ≤ M .

If is a module whose nonzero submodules are essential in , then is called a uniform module.

UU U






Iso result:

Every isoartinian module contains an isosimple submodule and so






Iso result:

Every isoartinian module contains an isosimple module and so

an essential submodule which is a (not necessarily finite) direct sum of isosimple modules.



As a consequence:

An integral domain is right isoartinian iff it is a PRID. D



Isonoetherian modules and uniform dimension.

Every noetherian module is of finite Goldie dimension.





Equivalently it contains an essential submodule which is a finite direct sum of uniform modules.





Iso case:





Iso case:

Every isonoetherian module is of finite Goldie dimension.





Iso case:


Every right isonoetherian domain is a right Ore domain.





Iso case:


Every right isonoetherian domain is a right Ore domain.

Open problem: Does every isonoetherian module have Krull dimension?



Suppose is a discrete valuation domain. R

If , then is a field. K . dim(R) = 0 R





If , then is a noetherian domain. K . dim(R) = 1 R






If , then is isonoetherian.K . dim(R) = 2 R






If , then is isonoetherian,K . dim(R) = 2 R but not noetherian.







The proof is based on this that a right chain domain is isonoetherian iff it satisfies ACC on infinitely generated right ideals.







If then is not isonoetherian. K . dim(R) ≥ 3, R



Endomorphism ring of an isosimple module:

The endomorphism ring of a simple module is a division ring.





Iso case: Let be endomorphism ring of an isosimple module . Then: E S






Every nonzero element of is injective. Ei)






Every nonzero element of is injective. E

is a right Ore domain. E

i)

ii)








satisfies the ascending chain condition on principal right ideals. E

i)

ii)

iii)








satisfies the ascending chain condition on principal right ideals. E

Open problem: Is a PRID?E

i)

ii)

iii)



a) is a PRID.E

b) E is a right Bezout domain.

c) is an intrinsically projective module. S

TFAE for the endomorphism ring of an isosimple module :E S



a) is a PRID.E



Right Bezout domain: every finitely generated right ideal is cyclic.




a) is a PRID.E



Intrinsically projective:

For every submodule of and any diagram

B S





a) is a PRID.E



Intrinsically projective:

For every submodule of and any diagram

B S, there is that makesh

the diagram commute.



Weddernburn Theorem:

A ring is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring.

R





R

So a simple ring is right artinian if and only if it is left artinian.

That is, of the form Matn(D) .





R



Iso result:





R



Iso result: A ring is simple and right isoartinian if and only if it is isomorphic to a Matrix ring over a PRID ring.

R





R



Iso result: A ring is simple and right isoartinian if and only if it is isomorphic to a Matrix ring over a PRID ring.

R

Remark:

It is not true that a simple ring is right isoartinian if and only if it left isoartinian.



SEMIPRIME RING:

A prime ideal in is an ideal such that whenever , then or P R xRy ⊆ P x ∈ P y ∈ P .



SEMIPRIME RING:


A semiprime ideal in is an ideal which is an intersection of prime ideals, P R



SEMIPRIME RING:


A semiprime ideal in is an ideal which is an intersection of prime ideals, or P R

equivalently due to Levitzki and Nagata, whenever implies xRx ⊆ P x ∈ P .



SEMIPRIME RING:




A prime ring is a ring whose zero ideal is a prime ideal.



SEMIPRIME RING:





A Semiprime ring is a ring whose zero ideal is a semiprime ideal.



SEMIPRIME RING:





A Semiprime ring is a ring whose zero ideal is a semiprime ideal. Equivalently,

there is no nilpotent ideal,



SEMIPRIME RING:





A Semiprime ring is a ring whose zero ideal is a semiprime ideal. Equivalently,

there is no nilpotent ideal, that is if , then I2 = 0 I = 0.



Jacobson radical and isoradical:

The Jacobson radical is the intersection of the annihilators of all simple right modules.





It is right/left symmetric notation.






Recall that annihilator of a module denoted by is ann is ann .

M (M)(M) = {r ∈ R |Mr = 0} It is a two sided ideal of R .






Iso case:

The right isoradical of a ring is the intersection of the annihilators of all isosimple right modules. We denote it - .

RI rad(R)






Iso case:

The right isoradical of a ring is the intersection of the annihilators of all isosimple right modules. We denote it - .

RI rad(R)

There exists a right noetherian right chain domain whose right isoradical and left isoradical are different.



Radical zero:

A right artinian ring is semiprime if and only if its radical is zero. R



Radical zero:


Iso case:

A right isoartinian ring is semiprime iff its isoradical is zero.R



Radical zero:


Iso case:


Proof: Let be right isoartinian, we show: R

is NOT semiprime if and only if - . R I rad(R) ≠ 0



Radical zero:


Iso case:




→



Radical zero:


Iso case:




Suppose that , for a nonzero ideal I2 = 0 I .→



Radical zero:


Iso case:




Suppose that , for a nonzero ideal I2 = 0 I .→If is an isosimple right module, we can see that . S SI = 0



Radical zero:


Iso case:





Otherwise, SI ≅ S and so which is a contradiction. 0 = SI2 ≅ SI ≅ S



Radical zero:


Iso case:





Therefore, - .I ⊆ I rad(R)



Radical zero:


Iso case:




←



Radical zero:


Iso case:




If - , then it is an isoartinian module and soI rad(R) ≠ 0←



Radical zero:


Iso case:




If - , then it is an isoartinian module and soI rad(R) ≠ 0

it contains an isosimple right ideal, say . C

←



Radical zero:


Iso case:




If - , then it is an isoartinian module and soI rad(R) ≠ 0

it contains an isosimple right ideal, say . C

←

Then - .C2 ⊆ C × I rad(R) = 0



TFAE for a ring :R

The Artin-Wedderburn Theorem: (Structure of semiprime right artinian rings)



a) is a semiprime right artinian ring. R

b) Every right -module is projective. R

c) is a direct sum of simple modules. RR

d) is isomorphic to a finite product of matrix rings over division rings. R


TFAE for a ring :R




a) is a semiprime right artinian ring. R

b) Every right -module is projective. R

c) is a direct sum of simple modules. RR

d) is isomorphic to a finite product of matrix rings over division rings. R

Iso case?

TFAE for a ring :R



TFAE for a semiprime right isoartinian ring : R




a) is right noetherian. R

b) is right isonoetherian. R

c) is of finite uniform dimension. RR






c)

is a direct sum of isosimple modules. RRd)

is sum of isosimple right ideals. R

is of finite uniform dimension. RR

e)

f) R is a finite product of matrix rings over PRIDs.






c)




e)


Open problem: Does satisfy the above equivalent condition!? R






c)




e)


Open problem: Does satisfy the above equivalent condition!? RThe answer isyes when is commutative.

R



Thanks for your attentions



A commutative domain is called Valuation ifD is is uniserial domain. DD

Equivalently it is a subring of a field and for every , either or belongs to .

Qq ∈ Q q q−1 D

Or equivalently there is a totally ordered abelian group and a filed and a surjetive group homomorphism such that

.

Γ Qv : Q× → Γ

D = {x ∈ Q |v(x) ≥ 0} ∪ {0}

Appendix:


Rings and modules with chain conditions up to isomorphism

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