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-1
votes
1answer
67 views

Direct sum of simple modules [on hold]

Let $N = M \oplus M$, where $M$ is a simple $R$-module. which one is correct (exactly one item)? 1- $N$ has finitely many sub-modules. 2- $Hom_R \left( {N,N} \right)$ is a division ring. 3- $Hom_R ...
1
vote
1answer
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 ...
2
votes
0answers
46 views

an equality in module theory

Let $R$ be a ring and $M$ be an $R$-module. Let $N$ and $L$ be two submodules of $M$ such that $(rad(N+L):M)=\sqrt{(N:M)+(L:M)}$. Then can we conclude that $(N+L:M)=(N:M)+(L:M)$? Note that ...
1
vote
0answers
185 views

R[[X]] flat as a R[X]-module?

I assume $R[X]\rightarrow R[[X]]$ is not flat in general, but I was wondering if any conditions on a commutative ring $R$ are known such that $R[[X]]$ is flat as a $R[X]$-module. Would $R$ noetherian ...
6
votes
0answers
86 views

Finiteness for separated residually finite modules

Suppose that $A$ is a commutative noetherian Jacobson ring and $M$ is an $A$-module. Suppose in addition that $M$ is $\mathfrak{m}$-adically separated for every maximal ideal $\mathfrak{m}$, and that ...
5
votes
1answer
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. ...
1
vote
2answers
239 views

A semisimple group ring

Let $n \in \mathbb{N}$, $p$ a prime number, and $G$ a finite group of order coprime to $p$. Let $R = \mathbb{Z} /p^n \mathbb{Z}$ be the ring of integers mod $p^n$. Must $R[G]$ be semisimple? As noted ...
3
votes
0answers
349 views

An exact sequence which does not split

Let $X$ and $Y$ be indecomposable modules over a finite dimensional algebra and let $f \colon X \to Y$ be a non-zero morphism which is neither a monomorphism nor an epimorphism. Suppose that it is ...
2
votes
1answer
54 views

dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...
-3
votes
1answer
212 views

Doubt in this proof of Horrocks theorem

I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...
11
votes
7answers
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 ...
7
votes
1answer
147 views

Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
1
vote
0answers
93 views

Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects? What are some applications? For ...
4
votes
1answer
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 ...
9
votes
1answer
516 views

Why are they called Specht Modules?

I know that the simple modules of $\mathbb{C}S_n$ are called $\it{Specht}$ $ \it{Modules}$, and they are named after the German Mathematician Wilhelm Specht because he studied them, but I think these ...
2
votes
1answer
130 views

Simultaneous decomposition of modules over Dedekind domains

I posted the question on mathexchange as well, but realized that my chances would be higher posting here; In the paper "Almost diagonal matrices over Dedekind Domains" by L. Levy a specific ...
1
vote
1answer
66 views

Injective modules over noncommutative noetherian rings

Let $R$ be a left noetherian ring, then it is well known that Matlis proved that any injective $R$-module decomposes into a sum of indecomposable injectives, each of form $E(R/I)$ for some irreducible ...
3
votes
1answer
260 views

Splitting as $\mathbb{F}_p[[X]]$-modules

Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB ...
4
votes
0answers
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
2answers
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
2answers
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
1answer
233 views

Is this square commutative?

Suppose that the following commutative diagram of $R$-modules is given. $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ ...
3
votes
1answer
257 views

Maximal Submodule of a Verma Module

Let $\mathfrak{h}$ be a Cartan subalgebra of a $\mathbb{C}$-semi simple Lie algebra $\mathfrak{g}$. Given $\lambda \in \mathfrak{h}^*$, $M(\lambda)$ the Verma module of highest weight $\lambda$ and ...
13
votes
5answers
724 views

is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
1
vote
1answer
128 views

arrows in the injective representations of quivers

Let $Q$ be a quiver (a directed graph) and $E$ be a representation of $Q$ by R-modules, that is, for each vertex $v$ in $Q$ there exists an R-module $E(v)$ and for each arrow $a:v\to w$ there exists a ...
5
votes
1answer
371 views

Splitness of commutative diagrams

Consider the following commutative diagram in the category of $R$-modules where $R$ is an associative ring with identity and all modules are unital. $ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ ...
1
vote
0answers
128 views

Is this a pure monomorphism?

Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is ...
1
vote
1answer
204 views

finitely presented representations

Let Q=(V,E) be a direct graph where V is the set of all its vertices and E denotes the set of all its arrows. $X$ is called a representation of Q by modules if it is a functor from Q to R-Mod. i.e. ...
1
vote
1answer
206 views

Pure monomorphism of functors-

Suppose that $F, G: Q\rightarrow R{\rm -Mod}$ be two covariant functors where $Q$ is an abelain category and $R$ is a commuatative ring. Also let $\eta: F\rightarrow G$ be a natural transformation ...
7
votes
1answer
239 views

Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I ...
1
vote
2answers
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
1answer
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
2answers
215 views

Looking for a reference: double orthogonal complement in $(\mathbb{Z}/q\mathbb{Z})^n$

I'm using the following result in a computer science paper: Let $V$ be a submodule of $(\mathbb{Z}/q\mathbb{Z})^n$ (n-tuples with addition and multiplication mod $q$). Let $$V^\perp = \{u \in ...
2
votes
3answers
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 ?
8
votes
3answers
445 views

What is the categorical significance of the trivial $\mathfrak{g}$-module in the category of $\mathfrak{g}$-mod?

This question may be trivial for experts. Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, ...
3
votes
0answers
225 views

Epimorphisms between external tensor products

Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k ...
2
votes
1answer
94 views

pure sub-complexes of exact subcomplexes

In https://www.google.com/#q=tensor+product+of+complexes%2Benochs a new tensor product of complexes is defined which characterizes flatness in the category of complexes of $R$-modules. That is, a ...
0
votes
1answer
171 views

cofree modules and dual

1, Why do people pay special attention to Q/Z in the definition of cofree modules instead of ordinary abelian groups? 2, Over a PID, is every injective module cofree? Just like the relationship ...
3
votes
1answer
133 views

is every finitely n-presented (S^{-1})R-module a localization of a finitely n-presented R-module?

Let S be a multiplicative set in a ring R. We can see that every finitely generated $(S^{-1})R$-module is a localization of a finitely generated R-module. Then, more generally, is every finitely ...
1
vote
1answer
165 views

Classification of pairs of commuting endomorphisms

Let $K$ be an algebraically closed field. I'm interested in isomorphism classes of triples $(V,f,g)$ where $V$ is a finite dimensional $K$-vector space and $f,g$ are commuting endomorphisms of $V$. ...
1
vote
0answers
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 ...
-1
votes
1answer
163 views

Pure Quotient and pure sub-object

Let $\mathcal{C}$ be the category of modules over a ring. Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ ...
3
votes
0answers
224 views

Rejects and injectives

Let $A$ be any ring and consider modules on the left. For $M$ $A$-module, the trace $Tr(M,A)$ is a two-sided ideal of $A$. If $A$ is a unitary ring then: $Tr(P,A)P=P$, for $P$ projective; ...
2
votes
1answer
82 views

Homocyclic primary module over PID

I posed the question here, but get no answers yet. Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime ...
20
votes
4answers
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
1answer
229 views

Maximal ideals and Essential ideals

(Edited) Both notions seem to measure "largeness" of an ideal in a ring. how are they related? does one imply the other and vice versa? i am reading this book "Hereditary Noetherian Prime Rings and ...
2
votes
1answer
505 views

finite length and finitely generated

it seems that a module over a noetherian ring R is finitely generated if and only if it has finite length (sorry, it turns out to be false! i must have had a misunderstanding!) but why in the ...
2
votes
1answer
154 views

Is there an $A$ such that $B$ injective iff 1st Ext functor vanishes?

In the category of $\mathbb{Z}$-modules, there exists a module $A$---for instance $\bigoplus_{k=2}^\infty \mathbb{Z}/k\mathbb{Z}$---such that a $\mathbb{Z}$-module $B$ is injective iff ...
0
votes
0answers
132 views

Finite rank free modules over PIDs

I have two finite rank free modules $M,N$ over a principal ideal domain (the ring of formal complex power series to be exact) and a homomorphism between these two modules $\phi:M\rightarrow N$. Under ...
-1
votes
2answers
309 views

Embedded associated prime and non zero divisor

$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is equidimensional and $M$ does not have any embedded prime. Given $x\in I$, where $I$ is an ideal of $A$, and $\dim ...