For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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1answer
27 views

$M_{1} \oplus M_{2}$ is a cyclic $A$-module $\iff \rm{Ann}(M_1)+\rm{Ann}(M_2)=A$ [duplicate]

Let $A$ be a commutative ring with an identity element $1$. An element $x$ in an $A$-module $M$ is called cyclic if $Ax=M$. An $A$-module which has a cyclic element is called cyclic $A$-module. Let $...
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0answers
15 views

Endowing module with algebra

Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have ...
0
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1answer
60 views

Same kernels for homomorphisms of free modules

Let $f: R^n \rightarrow R^m$ be an isomorphism of free $R$-modules ($R$ commutative with unity) and $\pi_1: R^n \rightarrow R^n/\mathfrak m^n$, $\pi_2: R^m \rightarrow R^m/\mathfrak m^m$ the canonical ...
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0answers
33 views

Let a,b have the same divisor (content) in an integral domain A. When can I deduce $a/b\in A^\times$?

Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be $div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P]$, where the sum ranges ...
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1answer
27 views

Module is isomorphic to a direct sum of modules, then the length(M)=sum length(M_i) [closed]

Is there any proof (or even a counterexample) for: If $M\tilde{=} M_1\oplus ... \oplus M_n$, then it follows for the finite length $l(M) = l(M_1)+...+l(M_n)$. (Modules of a commutative ring with ...
1
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1answer
34 views

If every cyclic $R$-module is projective, then $R$ is semisimple.

If every cyclic $R$-module is projective, then $R$ is semisimple. I know how to prove this if the definition of cyclic module is $M=Rm$ where $m$ is an element of $M$. The problem is, we defined in ...
0
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1answer
22 views

The existence of a non-split composition series in a indecomposable module

Assume that $R$ is a ring with unit and $M$ is a indecomposable left $R$-module with finite length. That is, $M$ has a composition series. Is it true that there is a composition series $$\begin{...
2
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1answer
23 views

Torsion in module over the ring of convergent power series

I need to understand a passage from a paper which I don't quite understand. Let $M$ be a module over the ring $\mathbb C\{t\}$ of convergent power series. We want to show that $M$ is torsion-free, i....
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0answers
9 views

for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\...
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0answers
14 views

Endomorphism of Central Simple Algebra

Let $A\in\mathscr{C}(F)$, i.e. $A$ is a central simple algebra. Show that $\text{End}_F(A)\cong M_n(F)$ as $F$-algebras where $n=\dim A$. My idea is to consider $A$ as a $F$-vector space, then $\...
0
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1answer
29 views

How can I prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?

I know that $\mathbb{Z}=2\mathbb{Z}+\{0\}$ (with "+" I mean "direct sum"). Is this enough to prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?
0
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1answer
20 views

Confused about order in Opposite Algebra

I am facing confusion in the "order of multiplication" regarding the opposite algebra $B^o$ in the following working: Define a right $B^o$-module $M_\phi$ where $M_\phi=M$ via $m\cdot b=m\phi(b)$. ...
2
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2answers
31 views

Elementary tensors

Let $G,H$ be $R$-modules, and $G \otimes H$ be it's tensor product. I can't prove it and I suspect it's false that any element $\tau \in G \otimes H$ can be written as $\tau = g \otimes h$ for some $...
2
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1answer
32 views

Regarding those $\mathbb{Z}$-modules whose every finite subset generates a finite submodule.

Let $X$ denote a $\mathbb{Z}$-module (aka an abelian group). Then $X$ may or may not satisfy: $(*)$ for all finite sets $F \subseteq X$, the module generated by $F$ is finite. This properly ...
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2answers
43 views

$M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ for cyclic modules $M,N$?

Is it true that $M \otimes_\mathbb{Z} N \cong \mathbb{Z}$ (considering $\mathbb{Z}$ as a $\mathbb{Z}$-module) if $M,N$ are cyclic $\mathbb{Z}$-modules (i. e. generated by one element)? I would say ...
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2answers
51 views

if $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ then $a_1 = a_2$

Assume $a_1 \cdot b \equiv a_2 \cdot b (\text{mod }n)$ and also $a_i^{\frac{n-1}{2}}$ mod $n \in \{1, n-1\}$ and $b^{\frac{n-1}{2}}$ mod $n \notin \{1, n-1\}$, I saw the following which proves $a_1 = ...
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0answers
35 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
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2answers
39 views

Showing $R\otimes M \cong M$ for $R$-modules $R,M$

How to see that $R \otimes M \cong M$ if $R$ and $M$ are $R$-modules (with $R$ being a commutative ring with unity)? I thought about defining $f: R\otimes M \rightarrow M$ by $(r,m) \mapsto rm$. ...
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2answers
18 views

Providing non trivial module morphisms

I'm starting to study modules, and I would like to get some counterexamples to naive ideas one has in the first approach to the subject. Does there exist an ideal I in A and a morphism $f:I \...
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0answers
89 views

Testing if a submodule is free

This is hopefully a very simple question. In Gröbner Bases in Commutative Algebra by Ene and Herzog, I find the Problem 4.11, which says ($S$ here is a polynomial ring over a field $K$, $S=K[x_1\ldots ...
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0answers
16 views

Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
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0answers
15 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...
1
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1answer
37 views

Tensor product (confusing question)

Let's say I have a tensor product $A\otimes B$ of algebras $A,B$. I have a linearly independent subset $S\subseteq A\otimes B$ such that $span(S)\cong C$, where $C$ is another algebra. Similarly, ...
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0answers
24 views

Composition series of a regular module.

Suppose $A$ is an $k$-algebra with basis ${1,e,s,t}$ and multiplication is given by $$ e^2 = e, es = s, te = t, s^2=t^2=se=et=st=ts=0. $$ I am trying to find the composition series for ...
2
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2answers
72 views

Corollary to Lemma of Nakayama

In Matsumura's Commutative Algebra there is the following Corollary to the Lemma of Nakayama: Let $A$ be a ring, $M$ an $A$-module, $N$ and $N'$ submodules of $M$, and $I$ an ideal of $A$. Suppose ...
2
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1answer
64 views

A better way of showing the set of all $m\times n$ matrix is an $R$ module.

Let $R$ be a ring and $m$ and $n$ be any positive integers. For any $a\in R$ and $A=(a_{ij})\in M_{m\times n}(R)$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}(R)$ is an $R$-module. Edited: ...
2
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1answer
37 views

Every module free implies no nonzero maximal ideal.

Show that given a ring $R$ with identity such that every $R$-module is free, then $R$ has no nonzero maximal ideals. I only know that every ideal of $R$ is a direct summand of $R$. Is it possible ...
0
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1answer
28 views

Show that $\operatorname{End}_A(P)$ is a central, simple $K$-algebra if $P$ is a f.g. projective $A$-module

Let $A$ be a central, simple $K$-algebra and let $P$ be a finitely generated projective $A$-module. I want to show that the endomorphism ring $\operatorname{End}_A(P)$ is also a central, simple $...
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3answers
40 views

why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$

This seems very simple and obvious but I can't prove it. why if $a \text{ mod } p = -1$ then $a \text{ mod } p = p-1$? thank you.
3
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3answers
87 views

Finite dimensional representations of the Weyl algebra in characteristic $p>0$

I'm working through representation theory course notes of P. Etingof. In problem 1.26 it is asked to find all finite dimensional irreducible representations of the algebra $A=\frac{k[x,y]}{\left\...
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3answers
36 views

Annihilator of modules [duplicate]

If $A$ is an $R$-module, I am having difficulty proving that $A$ is also a well-defined $R/ann(A)$-module with $(r+ann(A))a=ra$.
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21 views

A illuminating theorem on general modules over a ring

I'm going to do a one hour presentation on modules, free modules and projective modules over a ring. While most of my motivation for studying them comes from my interest in starting homological ...
2
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0answers
18 views

Injective homomorphism from an ideal to a torsion free module over a Dedekind ring

Let $M$ be a nonzero torsion free module over a Dedekind ring $R$ and $I ⊂ R$ an ideal. Show that there is an injective $R$-module homomorphism $I → M$. My solution is to take a nonzero element in ...
0
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1answer
14 views

Suppose $N \cong R^n $ then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\sum m_i \otimes e_i $

Suppose $N \cong R^n $ be free $R$ module of rank $n$ with basis $\{e_1,...,e_n\}$ and $R$ is commutative then for every non-zero $R$ module $M$ show that $M \otimes N$ can be written uniquely by $\...
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0answers
13 views

The module of infinite matrices has bases with any length, isn't it? [duplicate]

Let $R$ be a ring. We consider matrices of elements from $R$ with the following properties: The sizes of a matrix is infinite; Any row of a matrix have a finite number of nonzero elements of $R$ (...
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0answers
19 views

What is the rank of $\Bbb{Z}_m\oplus\Bbb{Z}_n$?

At first glance, the rank seems to be $2$. The basis elements seem to be $(0,1), (1,0)$. However, how is scalar multiplication defined? Is $a(p,q)=(ap,q)=(p,aq)$? So if the module under ...
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0answers
32 views

Proving “up to unique isomorphism”: Universal property of cokernel

This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism $f: M \...
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1answer
58 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from $...
1
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1answer
35 views

Submodules finitely generated if sum and intersection are finitely generated? [duplicate]

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine ...
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0answers
86 views

On describing a sort of “well-behaved” subgroups of a free abelian group.

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
2
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0answers
31 views

Any characterization of rings over which submodules of left free modules are free?

Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$? ...
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1answer
65 views

Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here $...
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2answers
48 views

Why does dual basis not span the dual space when the given vector space V is infinite dimensional?

We have a vector space V with basis $\mathcal{A}$. I have to show that the set $\mathcal{A}$* = { v* | v$\in\mathcal{A}$} does not span the dual space V*. I can not see why dual basis fails to span ...
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1answer
53 views

Lifting homomorphism when module is direct summand of free module [closed]

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module. Let further $f: M \rightarrow M''$ be a surjective homomorphism. I would like to show that for any homomorphism $\phi: N \...
1
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1answer
22 views

Question about proof of Krull principal ideal theorem

How can we explain the following step in the proof of Krull principal ideal theorem: $l\{ ((z):x^n)/(z) \}$ or $l\{ ((x^n):z)/(x^n) \}$ is finite? $l(M)$ - length of module.
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0answers
87 views

Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
1
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1answer
35 views

Applying hom-functor on short exact sequence

Let $N, M, M', M''$ be $R$-modules. Given homomorphisms $f: M' \rightarrow M$ $g: M \rightarrow M''$ $\psi: N \rightarrow M$ with $\operatorname{im} f = \ker g$, $g \circ \psi = 0$ and $g$ ...
1
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1answer
52 views

Natural action of $\mathbb{Z}G$ on $\mathbb{Z}$?

I'm studying projective modules and I'm having problem coming up with (or understanding) examples of non-free projective modules. I got that when a ring is a direct sum $R = A \oplus B$, both $A$ and $...
1
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0answers
34 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
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1answer
60 views

What exactly is the $O_X$-module and the corresponding sheaf of modules?

I am very puzzled by the definition in the Wiki page. I understand that over a subset $U$ we can assign a sheaf of abelian groups, e.g. some analytic functions over $U$. So we consider that these ...