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

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7
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179 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 ...
6
votes
0answers
78 views

Two ideals both alike in dignity, in fair Paris where we lay our scene.

Let $A$ be an integral domain. I have to show that two ideals $\mathfrak a$ and $\mathfrak b$ are isomorphic as $A$-modules if and only if there exist $a$ and $b$ such that $a\mathfrak b=b\mathfrak ...
6
votes
0answers
137 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 & ...
6
votes
0answers
254 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 ...
5
votes
0answers
65 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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0answers
95 views

Artinian rings are perfect

Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
5
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82 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
votes
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125 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 should ...
5
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221 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
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158 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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48 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
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67 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
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63 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace, I would ...
4
votes
0answers
46 views

$M\times N$ Doesn’t Have a Module Structure

In Keith Conrad's notes (page 4) is written: For two $R-$modules $M$ and $N$ , $M\oplus N$ and $M\times N$ are the same sets, but $M\oplus N$ is an $R-$module and $M\times N$ doesn’t have a ...
4
votes
0answers
82 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
votes
0answers
93 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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52 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
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139 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
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26 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
votes
0answers
73 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
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36 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
113 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
169 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
votes
0answers
113 views

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

I found this article about Galois Cohomology, http://www.math.uga.edu/~turkelli/Introduction%20to%20Galois%20Cohomology.pdf In it, it says that, when $G$ is a profinite group and $\mathbf{C}_G$ is ...
3
votes
0answers
33 views

On the Bass numbers of a local ring

Assume $R=k[x,y]/(x^2,xy,y^2)$, I would like to calculate the dimension as $k$-vectorspace of $\mathrm{Ext}^i_R(k,R)$. I see that as vector-space $\mathrm{Ext}^i_R(k,k)\cong k^{2^{i+1}}$, is it true ...
3
votes
0answers
30 views

Complements of semisimple rings

Let $R$ be a semisimple ring, let $M_{R}$ be a right-module then every right submodule of $M$ has a complement in $M$. Now assume $M$ is an $R$-bimodule with $R$ semisimple. Then every right ...
3
votes
0answers
45 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) ...
3
votes
0answers
47 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
84 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
57 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 ...
3
votes
0answers
104 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
47 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 ...
3
votes
0answers
66 views

$\text{Hom}$ of irreducible modules and restrictions

This question is in reference to this paper. More specifically it is in reference to the proof of proposition 1.4 on page 8. First a defintion: Let $A$ be a semisimple finite dimensional ...
3
votes
0answers
45 views

Projective Module as a Direct Sum of Left Ideals

I wonder if the following statement is true: Every projective $R$-module is a direct sum of projective left ideals of $R$. Most examples of non-free projective modules I have seen are all left ...
3
votes
0answers
64 views

Looking for a short book on modules

I'm looking for a short (100-150 pages) introductory book devoted to the algebra of modules. (I realize that many comprehensive algebra books include coverage of modules, but I'd prefer to read a ...
3
votes
0answers
32 views

Prove that $\eta: R/(a)\to R/(p_1^{\alpha_1})\oplus \cdots \oplus R/(p_r^{\alpha_r})$ is an epimorphism

Let $R$ be PID, and $a \in R$ with $a=p_1^{\alpha_1}\cdots p_r^{\alpha_r}$ its prime factorization. I'd like a Hint in proving that $$ x+(a) \to ...
3
votes
0answers
70 views

Help understanding a comment about isomorphisms of two direct sums.

I've recently noticed a few different posts asking about this problem Isomorphism of two direct sums, all using the CRT. I think I understand it this way. What caught my eye is in the above link a ...
3
votes
0answers
64 views

Free submodule of the intersection of free direct summands.

I have slightly more assumptions than the "standard" question (see e.g. this one) so the usual counterexamples that come to mind don't work (I believe). Let $R$ be a ring and $M$ a free $R$-module. ...
3
votes
0answers
305 views

Structure theorem of finitely generated modules over a PID

I want to prove the structure theorem of finitely generated modules over a PID using the primary decomposition in a Noetherian $R$-module $M$. Applying the results on primary decomposition to the ...
3
votes
0answers
93 views

On complexes of projective modules

How can I prove the following statement? Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then ...
3
votes
0answers
188 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 ...
3
votes
0answers
84 views

Computation of determinant of a matrix with elements from an arbitrary commutative ring

The cofactor formula for computing the determinant of a matrix is applicable when elements of the matrix are from a commutative ring. However, the complexity of this method is extremely high and I ...
3
votes
0answers
72 views

Conditions for equivalence of $A$-modules, given equivalence of $B$-modules for $B$ a subalgebra of $A$.

Pardon me if my terminology is messed up and I must admit that the following question is rather general, but here goes: Let $A$ be an algebra and $B$ a proper subalgebra of $A$. Suppose that, as ...
3
votes
0answers
66 views

Decompose a module $M$ of the form $N \times N$, where $N$ is simple

Let $M$ be a $\mathbb{C}[G]$-module of the form $M=N\times N$, where $N$ is simple. How to conclude that $M$ has infinitly many direct sum decompositions into two copies of $N$ ? This is what I have ...
3
votes
0answers
99 views

Question about ring and module

Consider a tangent bundle with even and odd parts $T_0 + T_1$, define a space $\Omega^{k,l}_{p,q}$ consisting of (p,q)-forms taking values in $\wedge^k T_0 \otimes \wedge^l T_1$, i.e. the space of ...
3
votes
0answers
192 views

group cohomology with coefficient in an induced module

We say that a $G$-module $I$ is induced if $$I\cong L\otimes\mathbb{Z}G$$ where $L$ is an abelian group and the action on $L\otimes\mathbb{Z}G$ is given by the action of $G$ only on the second ...
2
votes
0answers
53 views

If $\alpha$ and $\beta$ are algebraic integers then the roots of $x^2+\alpha x+\beta$ are algebraic integers

(This question is a dupplicate from If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.) I'm trying to solve this problem with ...
2
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0answers
8 views

Topological modules with enough continuous linear functionals.

Context: I'm trying to find out which topological (unital) modules are "good enough" for generalizing results from real or complex functional analysis. For example, I say that a module, in order to be ...
2
votes
0answers
79 views

Help in this proof in Lang's Algebra book (really elementary doubt)

My doubt is really elementary: Let $(f_1,\ldots,f_n)$ be a row matrix with some component with leading coefficient $1$ and $d$ the smallest degree of a component of $f$ with leading coefficient $1$. ...
2
votes
0answers
45 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 ...