# Tagged Questions

**1**

vote

**1**answer

110 views

### General criterion to find a Z-basis in a fixed generating subset

Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$
be a fixed finite subset.
Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the ...

**5**

votes

**1**answer

167 views

### Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such.
...

**11**

votes

**7**answers

684 views

### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

**4**

votes

**1**answer

228 views

### Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties:
(1) every finitely generated submodule of $M$ is projective ...

**4**

votes

**0**answers

92 views

### Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...

**0**

votes

**2**answers

279 views

### Dual of a module

Let $M$ be a $ \mathbb{Z}_{p}[[T]] $-module and $X=Hom(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ be the dual of $M$. Let $X[p^n]$ denotes the $p^n$-torsion points of $X$. Is $X/X[p^n]$ the dual of $M[p^n]$ ...

**13**

votes

**2**answers

388 views

### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...

**1**

vote

**2**answers

113 views

### On modules with finite uniform dimension

Is it true that if a module $M$ has finite uniform dimension then the same is true for its homomorphic images ?

**7**

votes

**1**answer

171 views

### Rings in which every module has an injective image

Consider the class of rings $R$ with identity such that any left $R$-module has a non-zero injective homomorphic image. Any such ring is clearly a left V-ring. Is it true that any such ring must be ...

**2**

votes

**3**answers

433 views

### Is it true that simple projective modules are injective?

It is known that simple modules are either projective or singular. Is it true that simple projective modules over (commutative) rings are injective ?

**1**

vote

**0**answers

124 views

### simple tensor product of modules over algebras

Let $M$, $N$ be simple modules over associative algebras $A$ and $B$ (over $\mathbb{C}$), respectively. When is $M\otimes N$ simple as a $A\otimes B$-module?
It is right if $A$ or $B$ has a ...

**20**

votes

**4**answers

1k views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**3**

votes

**3**answers

309 views

### Support of a module over a polynomial algebra

In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$
$$D \to E \to F,$$
we have that
Supp $E \subset$ ...

**1**

vote

**0**answers

205 views

### How to find the tensor product of modules that we don't know a basis for them?

Hi
I know how to calculate some easy tensor products like $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}\cong_{\mathbb{Z}} \mathbb{Z}/(m,n)\mathbb{Z} $ or $F[X] \otimes_{F} F[Y] ...

**1**

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**0**answers

199 views

### Average weighted value of a linear functional over increasing bounded subsets of Z^n

Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...

**4**

votes

**0**answers

322 views

### Kaplansky's theorem and Axiom of choice

Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...

**5**

votes

**2**answers

991 views

### Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. I'm seeking a kind of Schur's lemma, with $\mathrm{Hom}_R (M,N)$ replaced by $M \otimes_R N$. So my questions are:
Can ...

**4**

votes

**1**answer

283 views

### Projective dimension of simple module

Let $R$ be a (not necessarily commutative) ring and $M$ a simple right $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. It is seems known that
$$
...

**0**

votes

**1**answer

267 views

### Free Module with a Projective Sub- Module, and Tensor Products

Let us consider a unital algebra $A$, with a subalgebra $B \subseteq A$, along with an $A$-$A$-bimodule $M$ which is free as a right module, and a subspace $N$ (with respect to the action of the field ...

**0**

votes

**1**answer

229 views

### A Version of Nullstellensatz for Rings of Dİfferential Operators

Here is one of the classical versions of the nullstellensatz: Let $K$ be a field and let $\mathfrak{m}$ be a maximal ideal of the polynomial ring $K[T_1,\ldots,T_n]$. Then ...

**2**

votes

**1**answer

157 views

### Bimodule version of IBN

Hello all,
Does anyone have an example in mind of a ring $R$ for which $R^n\cong R^m$ as $R,R$ bimodules for some positive integers $n\neq m$?
I would be a little surprised if someone showed no such ...

**1**

vote

**0**answers

274 views

### Ring such that any submodule of an injective module is flat?

Does anyone know examples of rings $R$ with the property that any submodule of an injective (right) $R$-module is flat? If I'm not missing something, this class of rings includes the (Von Neumann) ...

**7**

votes

**3**answers

629 views

### Module category equivalent to graded module category?

Main Question
Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphisms that preserve ...

**1**

vote

**3**answers

912 views

### Generalization of eigenvalues/vectors to modules?

What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...

**3**

votes

**2**answers

329 views

### Modules of finite support

I'm reading Dwyer and Fried's paper "Homology of free abelian covers, I". In it, they make the following claim, which I'm having trouble verifying.
Let $F$ be a field and $A = F[x_1^{\pm ...

**3**

votes

**1**answer

221 views

### SBN and IBN rings

Hello, I can not figure out why a ring that is not IBN (invariant basis number) must be SBN (single basis number). More precisely: Let $R$ be a ring (with unit, generally non-commutative) such that ...

**6**

votes

**1**answer

404 views

### Are there non-reflexive modules isomorphic to their bi-dual?

Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism.
I'd like to know if there exists a module isomorphic to its bi-dual but not ...

**1**

vote

**1**answer

332 views

### Rank of a module

I have seen the definition of a module,not neccessary free, the alternatin sum of free modules in a free resolution of that module. it's clear that when the module is free our definition Coincide the ...

**5**

votes

**2**answers

669 views

### Lemma on infinitely generated projective modules

Is it true that every finitely generated submodule of a non-finitely generated projective over a (not necessarily commutative!) ring is contained in a proper summand?
N.B.: I asked this already on ...

**7**

votes

**2**answers

663 views

### Modules over Laurent series rings

Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...

**3**

votes

**1**answer

763 views

### structure theorem for modules

Can structure theorem for modules be extended to modules over UFDS , to modules over Neotherian rings ? if yes then can one get the statement and reference?
Since operations on matrices with ...

**2**

votes

**0**answers

162 views

### “Strongly finite rings” or similar term for a condition implying IBN

While (once again) perusing T.Y. Lam's excellent GTM 189 "Lectures on Modules and Rings" I compared the various conditions given that typically imply IBN, e.g. the (left) strong rank condition, stably ...

**4**

votes

**1**answer

1k views

### Are there any finitely generated artinian modules that are not notherian?

It is well known that for rings, Artinian implies Noetherian (the famous Hopkins–Levitzki theorem) and it is also well known that there are Artinian modules which are not Noetherian. A simple example ...

**16**

votes

**7**answers

1k views

### “Sums-compact” objects = f.g. objects in categories of modules?

Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...

**3**

votes

**2**answers

588 views

### What is the characteristic of the module over Jacobson semisimple ring？

We know a ring R is semisimple ring iff every module over R is semisimple，a ring R is von-Neumann regular ring iff every module over R is flat，What about the Jacobson semisimple ring？

**2**

votes

**3**answers

666 views

### Structure theorem for finitely generated Z[G] modules

For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?

**2**

votes

**0**answers

156 views

### Classification of finitely generated bimodules over “skew PID”s?

Let R be a noncommutative ring with a 1 and no zero divisors, such that all (two-sided) ideals of R are principally generated. Is there a classification theorem for finitely generated bimodules over ...

**3**

votes

**1**answer

242 views

### Behavior of the projective dimension of modules in a continuous chain of extensions

Let $R$ be an arbitrary ring. Let $D$ be the class of $R$-modules of projective dimension less than or equal to a natural number $n$. If $L$ is the direct union of a continuous chain of submodules ...

**10**

votes

**4**answers

578 views

### When are modules and representations not the same thing?

I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring ...

**1**

vote

**2**answers

424 views

### Understanding the modules of semiprimitive rings

As far as I understand, a semiprimitive ring can be fully 'explored' by its simple modules, in the sense that a semiprimitive ring is the subdirect product of its simple modules (for brevity, I'll use ...

**3**

votes

**0**answers

187 views

### Localization of power series and module structure

Let $R=\mathbb{Q}[X,Y]$ be the polynomial ring of two commuting variable.
Let $S$ be the multiplicative subset of $R$ generated by homogeneous linear polynomials.
Let also $\widehat{R}$ be the ring of ...