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. Noetherian ring: A ring R is called right noetherian if for every ascending chain of right ideals of , I1 ≤ I2 ≤ ⋯ R there exits an integer number n such that for every In = Ii i ≥ n .
Emmy Noether, 1882- 1935 Noetherian ring: A ring R is called right noetherian if for every ascending chain of right ideals of , I1 ≤ I2 ≤ ⋯ R there exits an integer number n such that for every In = Ii i ≥ n .
Emmy Noether, 1882- 1935 Noetherian modules: A module M is called noetherian if for every ascending chain of submodules of M1 ≤ M2 ≤ ⋯ M there exists such that n Mn = Mi for every i ≥ n . Artinian Modules: A module M is called artinian if for every descending chain of submodules M1 ≥ M2 ≥ ⋯ , there exists such that M n Mn = Mi for every Emil Artin , 1898 - i ≥ n . 1962 Artinian Modules: A module M is called artinian if for every descending chain of submodules M1 ≥ M2 ≥ ⋯ , there exists such that M n Mn = Mi for every Emil Artin , 1898 - i ≥ n . 1962
Every right Artinian ring is right Noetheran. 2016, Facchini and Nazemian, Isoartinian modules: 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if for every descending chain
of submodules of , M1 ≥ M2 ≥ ⋯ M 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if for every descending chain
of submodules of , M1 ≥ M2 ≥ ⋯ M there exists n such that
for every Mn ≅ Mi i ≥ n . 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if for every descending chain
of submodules of , M1 ≥ M2 ≥ ⋯ M there exists n such that
for every Mn ≅ Mi i ≥ n .
Right isoartinian ring: 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if for every descending chain
of submodules of , M1 ≥ M2 ≥ ⋯ M there exists n such that
for every Mn ≅ Mi i ≥ n .
If is an isoartinian module, the ring is called right Right isoartinian ring: RR R isoartinian. 2016, Facchini and Nazemian, Isoartinian modules:
A module M is called isoartinian if for every descending chain
of submodules of , M1 ≥ M2 ≥ ⋯ M there exists n such that
for every Mn ≅ Mi i ≥ n .
If is an isoartinian module, the ring is called right Right isoartinian ring: RR R isoartinian.
You can define left isoartinian similarly. Also talk about isoartinian ring. 2016, Facchini and Nazemian, isonoetherian modules:
A module M is called isonoetherian if for every ascending chain of submodules of , there exists such M1 ≤ M2 ≤ ⋯ M n that for every Mn ≅ Mi i ≥ n . 2016, Facchini and Nazemian, isonoetherian modules:
A module M is called isonoetherian if for every ascending chain of submodules of , there exists such M1 ≤ M2 ≤ ⋯ M n that for every Mn ≅ Mi i ≥ n .
Right (left) If ( ) is isonoetherian module, the ring is called RR RR R isonoetherian ring: right (left) isonoetherian.
Also we can talk about isonoetherian ring. Isosimple modules:
Recall that a nonzero module M is called simple if it has no nonzero proper submodule.
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Recall that a nonzero module M is called simple if it has no nonzero proper submodule.
We call a nonzero module M isosimple if every nonzero submodule of M is isomorphic to M .
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Recall that a nonzero module M is called simple if it has no nonzero proper submodule.
We call a nonzero module M isosimple if every nonzero submodule of M is isomorphic to M .
Every submodule in an isosimple module is cyclic.
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Recall that a nonzero module M is called simple if it has no nonzero proper submodule.
We call a nonzero module M isosimple if every nonzero submodule of M is isomorphic to M .
Every submodule in an isosimple module is cyclic.
If is isosimple, where is a domain, then is a PRID. DD D D
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Consider the abelian group ℤ.
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Consider the abelian group ℤ.
All nonzero submodules of ℤ are isomorphic to ℤ, that is, every submodule N of ℤ is in form mℤ for some integer m. We know mℤ ≅ ℤ. So ℤ is isoartinian, isosimple and isonoetherian.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
It is always isoartinian even if it is of infinite dimension.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
It is always isoartinian even if it is of infinite dimension.
Indeed if is a descending chain of submodules, then V1 ≥ V2 ≥ ⋯ we
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
It is always isoartinian even if it is of infinite dimension.
Indeed if is a descending chain of submodules, then V1 ≥ V2 ≥ ⋯ have a descending chain of cardinals we dim(V1) ≥ dim(V2) ≥ ⋯,
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
It is always isoartinian even if it is of infinite dimension.
Indeed if is a descending chain of submodules, then V1 ≥ V2 ≥ ⋯ we have a descending chain of cardinals dim(V1) ≥ dim(V2) ≥ ⋯, we know that every descending chain of cardinals stops somewhere, and two vector spaces over k are isomorphic if and only if they are of the same dimension.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
It is isonoetherian if and only if it is of finite length.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
Free modules over a PID (principal ideal domain).
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
Free modules over a PID (principal ideal domain).
k Vk If Vk is an infinite dimensional vector space, the ring is a (0 k ) right isoartinian ring which is not right isonoetherian.
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Consider the abelian group ℤ. A PRID is isosimple, isonoetherian and isoartinian.
Consider any vector space Vk , where k is field.
Free modules over a PID (principal ideal domain).
k Vk If Vk is an infinite dimensional vector space, the ring is a (0 k ) right isoartinian ring which is not right isonoetherian.
In the following, I will give you more examples….
Visit http://keynotetemplate.com for more free keynote resources! Artinian modules are essential extensions of their socle.
Visit http://keynotetemplate.com for more free keynote resources! Artinian modules are essential extensions of their socle.
Every artinian module contains a simple submodule.
Visit http://keynotetemplate.com for more free keynote resources! 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.
Visit http://keynotetemplate.com for more free keynote resources! 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 0 ≠ N ≤ M is called essential in M if
Visit http://keynotetemplate.com for more free keynote resources! 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 0 ≠ N ≤ M is called essential in M if
N ∩ N′ ≠ 0 for every nonzero N′ ≤ M .
Visit http://keynotetemplate.com for more free keynote resources! 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 0 ≠ N ≤ M is called essential in M if
N ∩ N′ ≠ 0 for every nonzero N′ ≤ M .
If U is a module whose nonzero submodules are essential in U, then U is called a uniform module.
Visit http://keynotetemplate.com for more free keynote resources! 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.
Iso result:
Every isoartinian module contains an isosimple submodule and so
Visit http://keynotetemplate.com for more free keynote resources! 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.
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.
Visit http://keynotetemplate.com for more free keynote resources! As a consequence:
An integral domain D is right isoartinian iff it is a PRID.
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Every noetherian module is of finite Goldie dimension.
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Every noetherian module is of finite Goldie dimension.
Equivalently it contains an essential submodule which is a finite direct sum of uniform modules.
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Every noetherian module is of finite Goldie dimension.
Iso case:
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Every noetherian module is of finite Goldie dimension.
Iso case:
Every isonoetherian module is of finite Goldie dimension.
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Every noetherian module is of finite Goldie dimension.
Iso case:
Every isonoetherian module is of finite Goldie dimension.
Every right isonoetherian domain is a right Ore domain.
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Every noetherian module is of finite Goldie dimension.
Iso case:
Every isonoetherian module is of finite Goldie dimension.
Every right isonoetherian domain is a right Ore domain.
Open problem: Does every isonoetherian module have Krull dimension?
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
If K . dim(R) = 1, then R is a noetherian domain.
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
If K . dim(R) = 1, then R is a noetherian domain.
If K . dim(R) = 2, then R is isonoetherian.
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
If K . dim(R) = 1, then R is a noetherian domain.
If K . dim(R) = 2, then R is isonoetherian, but not noetherian.
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
If K . dim(R) = 1, then R is a noetherian domain.
If K . dim(R) = 2, then R is isonoetherian, but not noetherian.
Te proof is based on tis tat a right chain domain is isonoeterian iff it satsfies ACC on infinitly generatd right ideals.
Visit http://keynotetemplate.com for more free keynote resources! Suppose R is a discrete valuation domain.
If K . dim(R) = 0, then R is a field.
If K . dim(R) = 1, then R is a noetherian domain.
If K . dim(R) = 2, then R is isonoetherian, but not noetherian.
If K . dim(R) ≥ 3, then R is not isonoetherian.
Visit http://keynotetemplate.com for more free keynote resources! Endomorphism ring of an isosimple module:
The endomorphism ring of a simple module is a division ring.
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The endomorphism ring of a simple module is a division ring.
Iso case: Let E be endomorphism ring of an isosimple module S. Then:
Visit http://keynotetemplate.com for more free keynote resources! Endomorphism ring of an isosimple module:
The endomorphism ring of a simple module is a division ring.
Iso case: Let E be endomorphism ring of an isosimple module S. Then:
i) Every nonzero element of E is injective.
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The endomorphism ring of a simple module is a division ring.
Iso case: Let E be endomorphism ring of an isosimple module S. Then:
i) Every nonzero element of E is injective.
ii) E is a right Ore domain.
Visit http://keynotetemplate.com for more free keynote resources! Endomorphism ring of an isosimple module:
The endomorphism ring of a simple module is a division ring.
Iso case: Let E be endomorphism ring of an isosimple module S. Then:
i) Every nonzero element of E is injective.
ii) E is a right Ore domain.
iii) E satisfies the ascending chain condition on principal right ideals.
Visit http://keynotetemplate.com for more free keynote resources! Endomorphism ring of an isosimple module:
The endomorphism ring of a simple module is a division ring.
Iso case: Let E be endomorphism ring of an isosimple module S. Then:
i) Every nonzero element of E is injective.
ii) E is a right Ore domain.
iii) E satisfies the ascending chain condition on principal right ideals.
Open problem: Is E a PRID?
Visit http://keynotetemplate.com for more free keynote resources! TFAE for the endomorphism ring E of an isosimple module S:
a) E is a PRID.
b) E is a right Bezout domain.
c) S is an intrinsically projective module.
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a) E is a PRID.
b) E is a right Bezout domain.
c) S is an intrinsically projective module.
Right Bezout domain: every finitely generated right ideal is cyclic.
Visit http://keynotetemplate.com for more free keynote resources! TFAE for the endomorphism ring E of an isosimple module S:
a) E is a PRID.
b) E is a right Bezout domain.
c) S is an intrinsically projective module.
Intrinsically projective:
For every submodule B of S and any diagram
Visit http://keynotetemplate.com for more free keynote resources! TFAE for the endomorphism ring E of an isosimple module S:
a) E is a PRID.
b) E is a right Bezout domain.
c) S is an intrinsically projective module.
Intrinsically projective:
For every submodule B of S and any diagram , there is h that makes the diagram commute.
Visit http://keynotetemplate.com for more free keynote resources! Weddernburn Theorem:
A ring R is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring.
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A ring R is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring. That is, of the form Matn(D) .
So a simple ring is right artinian if and only if it is left artinian.
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A ring R is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring. That is, of the form Matn(D) .
So a simple ring is right artinian if and only if it is left artinian.
Iso result:
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A ring R is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring. That is, of the form Matn(D) .
So a simple ring is right artinian if and only if it is left artinian.
Iso result: A ring R is simple and right isoartinian if and only if it is isomorphic to a Matrix ring over a PRID ring.
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A ring R is simple and right artinian if and only if it is isomorphic to a matrix ring over a division ring. That is, of the form Matn(D) .
So a simple ring is right artinian if and only if it is left artinian.
Iso result: A ring R is simple and right isoartinian if and only if it is isomorphic to a Matrix ring over a PRID ring.
Remark:
It is not true that a simple ring is right isoartinian if and only if it left isoartinian.
Visit http://keynotetemplate.com for more free keynote resources! SEMIPRIME RING:
A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
Visit http://keynotetemplate.com for more free keynote resources! SEMIPRIME RING:
A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals,
Visit http://keynotetemplate.com for more free keynote resources! SEMIPRIME RING:
A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals, or
equivalently due to Levitzki and Nagata, whenever xRx ⊆ P implies x ∈ P .
Visit http://keynotetemplate.com for more free keynote resources! SEMIPRIME RING:
A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals, or
equivalently due to Levitzki and Nagata, whenever xRx ⊆ P implies x ∈ P .
A prime ring is a ring whose zero ideal is a prime ideal.
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A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals, or
equivalently due to Levitzki and Nagata, whenever xRx ⊆ P implies x ∈ P .
A prime ring is a ring whose zero ideal is a prime ideal.
A Semiprime ring is a ring whose zero ideal is a semiprime ideal.
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A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals, or
equivalently due to Levitzki and Nagata, whenever xRx ⊆ P implies x ∈ P .
A prime ring is a ring whose zero ideal is a prime ideal.
A Semiprime ring is a ring whose zero ideal is a semiprime ideal. Equivalently,
there is no nilpotent ideal,
Visit http://keynotetemplate.com for more free keynote resources! SEMIPRIME RING:
A prime ideal P in R is an ideal such that whenever xRy ⊆ P, then x ∈ P or y ∈ P .
A semiprime ideal P in R is an ideal which is an intersection of prime ideals, or
equivalently due to Levitzki and Nagata, whenever xRx ⊆ P implies x ∈ P .
A prime ring is a ring whose zero ideal is a prime ideal.
A Semiprime ring is a ring whose zero ideal is a semiprime ideal. Equivalently,
there is no nilpotent ideal, that is if I2 = 0, then I = 0.
Visit http://keynotetemplate.com for more free keynote resources! Jacobson radical and isoradical:
The Jacobson radical is the intersection of the annihilators of all simple right modules.
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The Jacobson radical is the intersection of the annihilators of all simple right modules.
It is right/left symmetric notation.
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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 M denoted by is ann(M) is ann(M) = {r ∈ R|Mr = 0}. It is a two sided ideal of R .
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The Jacobson radical is the intersection of the annihilators of all simple right modules.
It is right/left symmetric notation.
Iso case:
The right isoradical of a ring R is the intersection of the annihilators of all isosimple right modules. We denote it I-rad(R).
Visit http://keynotetemplate.com for more free keynote resources! Jacobson radical and isoradical:
The Jacobson radical is the intersection of the annihilators of all simple right modules.
It is right/left symmetric notation.
Iso case:
The right isoradical of a ring R is the intersection of the annihilators of all isosimple right modules. We denote it I-rad(R).
There exists a right noetherian right chain domain whose right isoradical and left isoradical are different.
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A right artinian ring R is semiprime if and only if its radical is zero.
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A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
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A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0.
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. →
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. → Suppose that I2 = 0, for a nonzero ideal I .
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. → Suppose that I2 = 0, for a nonzero ideal I . If S is an isosimple right module, we can see that SI = 0.
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. → Suppose that I2 = 0, for a nonzero ideal I . If S is an isosimple right module, we can see that SI = 0.
Otherwise, SI ≅ S and so 0 = SI2 ≅ SI ≅ S which is a contradiction.
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. → Suppose that I2 = 0, for a nonzero ideal I . If S is an isosimple right module, we can see that SI = 0.
Therefore, I ⊆ I-rad(R).
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. ←
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. ← If I-rad(R) ≠ 0, then it is an isoartinian module and so
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. ← If I-rad(R) ≠ 0, then it is an isoartinian module and so it contains an isosimple right ideal, say C.
Visit http://keynotetemplate.com for more free keynote resources! Radical zero:
A right artinian ring R is semiprime if and only if its radical is zero.
Iso case:
A right isoartinian ring R is semiprime iff its isoradical is zero.
Proof: Let R be right isoartinian, we show: R is NOT semiprime if and only if I-rad(R) ≠ 0. ← If I-rad(R) ≠ 0, then it is an isoartinian module and so it contains an isosimple right ideal, say C.
Then C2 ⊆ C × I-rad(R) = 0.
Visit http://keynotetemplate.com for more free keynote resources! The Artin-Wedderburn Theorem: (Structure of semiprime right artinian rings)
TFAE for a ring R:
Visit http://keynotetemplate.com for more free keynote resources! The Artin-Wedderburn Theorem: (Structure of semiprime right artinian rings)
TFAE for a ring R:
a) R is a semiprime right artinian ring.
b) Every right R-module is projective.
c) is a direct sum of simple modules. RR
d) R is isomorphic to a finite product of matrix rings over division rings.
Visit http://keynotetemplate.com for more free keynote resources! The Artin-Wedderburn Theorem: (Structure of semiprime right artinian rings)
TFAE for a ring R:
a) R is a semiprime right artinian ring.
b) Every right R-module is projective.
c) is a direct sum of simple modules. RR
d) R is isomorphic to a finite product of matrix rings over division rings.
Iso case?
Visit http://keynotetemplate.com for more free keynote resources! TFAE for a semiprime right isoartinian ring R:
Visit http://keynotetemplate.com for more free keynote resources! TFAE for a semiprime right isoartinian ring R:
a) R is right noetherian.
b) R is right isonoetherian.
c) is of finite uniform dimension. RR
Visit http://keynotetemplate.com for more free keynote resources! TFAE for a semiprime right isoartinian ring R:
a) R is right noetherian.
b) R is right isonoetherian.
c) is of finite uniform dimension. RR
d) is a direct sum of isosimple modules. RR
e) R is sum of isosimple right ideals.
f) R is a finite product of matrix rings over PRIDs.
Visit http://keynotetemplate.com for more free keynote resources! TFAE for a semiprime right isoartinian ring R:
a) R is right noetherian.
b) R is right isonoetherian.
c) is of finite uniform dimension. RR
d) is a direct sum of isosimple modules. RR
e) R is sum of isosimple right ideals.
f) R is a finite product of matrix rings over PRIDs.
Open problem: Does R satisfy the above equivalent condition!?
Visit http://keynotetemplate.com for more free keynote resources! TFAE for a semiprime right isoartinian ring R:
a) R is right noetherian.
b) R is right isonoetherian.
c) is of finite uniform dimension. RR
d) is a direct sum of isosimple modules. RR
e) R is sum of isosimple right ideals.
f) R is a finite product of matrix rings over PRIDs.
The answer is Open problem: Does R satisfy the above equivalent condition!? yes when R is commutative. Visit http://keynotetemplate.com for more free keynote resources! Tanks for your atentons
Visit http://keynotetemplate.com for more free keynote resources! Appendix:
A commutative domain is called Valuation if is is uniserial domain. D DD
Equivalently it is a subring of a field Q and for every q ∈ Q, either q or q−1 belongs to D. Or equivalently there is a totally ordered abelian group Γ and a filed Q and a surjetive group homomorphism v : Q× → Γ such that D = {x ∈ Q|v(x) ≥ 0} ∪ {0}.
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