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

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16 views

Testing if a submodule is free

This is hopefully a very simple question. In "Groebner 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 $K$, ...
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0answers
13 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
14 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 ...
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1answer
36 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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20 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 ...
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2answers
45 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 ...
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0answers
24 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 $A=(a_{ij})\in M_{m\times n}$ define $aA=(aa_{ij})$. Then prove that $M_{m\times n}$ is an $R$-module. Edited: Take ...
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1answer
33 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 ...
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1answer
22 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
38 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.
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2answers
19 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 ...
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0answers
16 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 ...
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0answers
16 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 ...
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1answer
11 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
9 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
18 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
25 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
53 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 ...
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1answer
26 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
28 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
58 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
38 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 ...
2
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1answer
36 views

Torsion-free divisible module over a commutative integral domain is injective. [duplicate]

This question is from the book Basic Algebra by P.M. Cohn. Show that a torsion free divisible module over a commutative integral domain is injective.
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1answer
23 views

Lifting homomorphism when module is direct summand of free module

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 ...
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1answer
20 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
53 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 ...
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1answer
23 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 $g$ ...
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28 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 ...
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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
50 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 ...
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1answer
39 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if ...
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0answers
9 views

Submodules of direct sum of simple modules [duplicate]

Is it true that if $N$ is a simple module, then the only non trivial submodule of $N\oplus N$ is $N$? I am aware of the "diagonal" submodule $\{(x,x)\mid x\in N\}$, but that is isomorphic to $N$?
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1answer
30 views

Abelian group structure and structure of Z[i]

Let $F = \Bbb{Z_p}$. For which prime integers $p$ does the additive group $F^1$ have a structure of $\Bbb{Z}[i]$-module? How about for $F^2$? I'm not really sure on how to approach this question, do ...
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0answers
66 views

Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of ...
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1answer
32 views

Showing that $A^n/M^n \cong (A/M)^n$

I am just trying to show that for $M$ a maximal ideal we have an isomorphism of $A$-modules $A^n / M^n \cong (A / M)^n$. I wanted to make sure that I have correctly understood the concepts and what ...
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17 views

Showing F-matrix representation is irreducible over $\mathbb{R}$

I have $G$ the cyclic group of order 4 and its $F$-(matrix) representation $T$ is $$\hat{T}(g) = \bigg[ \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \bigg]. $$ I am trying to show that ...
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2answers
38 views

Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated ...
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0answers
8 views

Wedderburn rings, Projective Modules

Prove that an artinian ring with no nonzero nilpotent elements and no non trivial central idempotents is a division ring.
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1answer
36 views

For every ring R, every left R-module M can be imbedded as a submodule of an injective left R-module [closed]

Prove that for every ring $R$, every left $R$-module $M$ can be imbedded as a submodule of an injective left $R$-module
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16 views

Maximal $\mathbb Z_p$-orders in $\mathbb Q_p[G]$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. ...
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1answer
23 views

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321

Use the prime factorization of $\phi(321)$ to determine possible orders of units mod 321. (Your list should have fewer than ten numbers in it). My attempt: $\phi(321)$ = $\phi(3)\phi(107)$ = $2(3)$ ...
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0answers
26 views

Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any ...
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2answers
26 views

Generating system for $R$-module $M$ still generating $\mathfrak aM$?

Let $M$ be a module over a commutative ring $R$ with unity and $\mathfrak a$ an ideal in $R$. Let $m_1, ..., m_n \in M$ be a generating system for $M$ and $z \in\mathfrak a M$. Then there are $a_1, ...
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1answer
33 views

Module over an infinite dimensional algebra

I have two question related to infinite dimensional algebra I have been seen a lot of example about Module over a finite dimensional $k-$algebra, but I could not find a literature about Module ...
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2answers
26 views

Dimension of a basis for a module using the quotient map

I'm going through a proof that the size of a basis for a module is uniquely determined and can't see why the following line from my lecture notes follows: "We now claim that if $X$ is a basis for ...
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0answers
206 views

Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
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1answer
29 views

What is the cokernel of the map $A:\mathbb Z^2\to \mathbb Z^7$?

Consider the free $\mathbb Z$ - modules $\mathbb Z^2$ and $\mathbb Z^7$ and consider the $\mathbb Z$ - module map $A:\mathbb Z^2\to \mathbb Z^7$ given by, ...
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1answer
23 views

Support of a module over polynomials: intuitive meaning

I find myself working with modules over complex polynomials in more variables, say $C[u_1,\dots,u_l]$. One can introduce the concept of support of the module $H$ as \begin{equation} supp H = ...
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7 views

Equivalence between absolutely pure modules and FP-injective modules

Let $R$ b a ring and $Q_{R}$ a right $R$-module. Show that the following conditions are equivalent for $Q_{R}$ : $(i)$ $Q_{R}$ is FP-injective, that is, given any short exact sequence $0 \rightarrow ...
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1answer
22 views

Find a vector $v$ such that $V=\mathbb{C}[T]\cdot v.$

Let $V=\mathbb{C}^2$ and let $\alpha:V\to V$ be the $\mathbb{C}$-linear map given by \begin{equation*} \begin{pmatrix} a\\ b \end{pmatrix} \longmapsto \begin{pmatrix} 2 & 3\\ -3 & 8 ...