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

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10
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218 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
9
votes
0answers
42 views

Why are noetherian and artinian modules important?

As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem ...
9
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237 views

An elegant description for graded-module morphisms with non-zero zero component

In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
7
votes
0answers
29 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
7
votes
0answers
316 views

Condition for a ring on projective and free modules problem

Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
6
votes
0answers
76 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to ...
6
votes
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264 views

Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem

For any matrix A with entries in a PID, there exist invertible matrices P and Q such that B = PAQ, where B is in Smith normal form. This theorem is usually proved by using elementary row/column ...
5
votes
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39 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
5
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163 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & ...
5
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94 views

Category of Chain Complexes of $R$-modules

So I have a couple of questions: 1- Formally speaking, what is a "quotient of a chain complex" of $R$-modules? 2- I want to show that any chain complexes of $R$-modules $C_\bullet $ is the ...
5
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93 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
5
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178 views

Dual modules and first cohomology

Let $G$ be a finite group, $K$ a characteristic-$p$ algebraically closed field (say $p$ divides $|G|$), and let $M$ be a finite-dimensional $KG$-module. What hypotheses are needed on $G$, $M$ to ...
4
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70 views

Property of free submodules for a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=<x>$ , ...
4
votes
0answers
130 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and ...
4
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0answers
41 views

On semisimple rings

Let $R$ denote a ring with unity. I know that, if $R$ is semisimple, then every $R$-module is semisimple. In particular the class of indecomposable $R$-modules coincides with the class of simple ...
4
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46 views

Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
4
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62 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
4
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65 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
4
votes
0answers
127 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
4
votes
0answers
66 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
4
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111 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...
4
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67 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
4
votes
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115 views

Finitely presented modules and torsion

Suppose $R$ is a Dedekind domain and $A$ a $(m \times n)$-matrix of rank $r$ over $R$. $A$ induces an $R$-module homomorphism $\varphi:R^m \to R^n$ via $x \mapsto xA$ giving rise to an exact sequence ...
4
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68 views

Decomposing the group algebra $\mathbb{R}[Q]$ as a product of matrix rings

I am trying to decompose $\mathbb{R}[Q]$ as a product of matrix rings ($Q$ is the group of quaternions) By Maschke's theorem, $\mathbb{R}[Q]$ is semisimple. I will begin by decomposing it as a left ...
4
votes
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157 views

Comparison of positive elements and Hilbert C*-modules

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras. Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
4
votes
0answers
28 views

An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
4
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0answers
77 views

Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
4
votes
0answers
41 views

Normal Form problem (in a module over a PID)

Let $A,B$ be $n\times n$ matrices with entires in a PID $D$ and $\det AB\neq 0$. Suppose diag$\{a_i\}$, diag$\{b_i\}$, and diag$\{c_i\}$ are normal froms for $A$, $B$, and $AB$. In particular, ...
4
votes
0answers
321 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
4
votes
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119 views

Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
4
votes
0answers
241 views

Zero divisors in modules?

Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that $\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ . For what kinds of ...
4
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0answers
127 views

Why has the category of all discrete $G$-modules not enough projectives when $G$ is profinite?

I found this article about Galois Cohomology. In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is the category of all discrete $G$-modules, $\mathbf{C}_G$ doesn't have enough ...
3
votes
0answers
31 views

What would be an example such that cardinality of linearly independent set is greater than the rank?

Let $R$ be a commutative ring and $M$ be a free $R$-module. Since $R$ is commutative, $R$ has IBN property, hence the rank of $M$ is uniquely well-defined. So set $n:=rnk(M)$. Let $A$ be an ...
3
votes
0answers
38 views

For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?

Let $X$ and $Y$ be abelian groups. Suppose $\text{Hom}(X,Z)\cong \text{Hom}(Y,Z)$ for all abelian groups $Z$. Does it follow that $X \cong Y$? It has been answered before that this is true if the ...
3
votes
0answers
52 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
3
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0answers
56 views

finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
3
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0answers
42 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
3
votes
0answers
83 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
3
votes
0answers
84 views

$\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff f_I(M)\subseteq f_I(N) $?

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. It has proven (here) that if $\operatorname{supp}(M) \subseteq \operatorname{supp}(N)$ then ...
3
votes
0answers
95 views

Length of a composition series of a module

If $A=\mathbb{C}[x,y]_{(x,y)}$, then what is the length of $A$-module $$A/(x^3-x^2y^2+y^{100},x^3-y^{999})\ ?$$ Any suggestion ?
3
votes
0answers
67 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
3
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0answers
77 views

When is $N\otimes_A B \to N$ an isomorphism?

Let $A, B$ be commutative (unital) rings and $f\colon A \to B$ an $A$-algebra. There then exists a canonical functor $f_*\colon \mathbf{Mod}_B \to \mathbf{Mod}_A$ such that, for every morphism of ...
3
votes
0answers
86 views

Projective and simple modules over finite dimensional algebras

I'm working through some lecture notes on the representation theory of a finite dimensional algebra $A$ (associative, unital, over an algebraically closed field $k$), and have got stuck on a ...
3
votes
0answers
70 views

Approximating modules over complete local rings

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto ...
3
votes
0answers
61 views

proving an isomorphism of direct limits

Let $(R,m,k)$ be a local Noetherian ring and $M$ an $R$-module. Let $\left\{I_s\right\}_s$ be a directed system of ideals whose induced topology is equivalent to the $m$-adic topology. Using the ...
3
votes
0answers
78 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
3
votes
0answers
57 views

“Localization” of a module at a family of elements

Let $x=(x_i)_{i \in I}$ be a family of elements of a commutative ring $R$. Typically $I$ is infinite. Let $M$ be an $R$-module. For every finite subset $E \subseteq I$ define $M_E = M$, and for ...
3
votes
0answers
101 views

Question from Liu, Chapter 5 Ex 1.16

I have a question from Liu's book on Algebraic Geometry, doing Chapter 5 Question 1.16: Let $f:X\rightarrow S$ be a morphism of schemes, with the following base change. $$\require{AMScd} \begin{CD} ...
3
votes
0answers
116 views

R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
3
votes
0answers
49 views

When does $\hom(A,-)$ have trivial kernel?

If $R$ is a commutative ring, is there a special name for those $R$-modules $A$ with the following property? $\forall R$-modules $M$: $~\hom(A,M) = 0 \Rightarrow M = 0$ Notice that every every ...