Questions tagged [modules]

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

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Extending epimorphisms from subspaces.

Suppose $U$ and $V$ are finite-dimensional vector spaces over some field $\mathbb{K}$, and $W$ is a subspace of $V$. We know that any homomorphism $f : U \to W$ can be extended to a homomorphism $f^* :...
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A question about split exact sequence in rng category

For $R$-module category, we have the notion of split exact sequence which is given by the following. For a fixed ring $R$, let $M,M^\prime,M^{\prime\prime}$ be modules over $R$. We call an exact ...
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Relationship between $M$ as an $A$-module and as a $B$-module, where $B=\operatorname{End}_A(M)$ [closed]

Let $A$ be a $K$-algebra, $M$ a left $A$-module and $B=\operatorname{End}_A(M)$. Show that: a) If $M$ is a generator as a left $A$-module, then it is projective as a right $B$-module. b) If $M$ is ...
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Computing local cohomology over a noncommutative ring

Let $R$ be an $\mathbb{N}$-graded ring, and let $\Gamma : R\text{-gr.mod} \to R\text{-gr.mod}$ be the left exact functor defined by $$ \Gamma(M) = \{ m \in M \mid R_{\geq n} m = 0 \text{ for some } n \...
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Smith Normal forms in a polynomial ring

Let $M$ be a $\mathbb{C[x]}$ module with generators $m_1,m_2$ and relations : $$(x^2+ix)m_1+(x+i)m_2=0 \\ (-2x+2i)m_1+ (x^2+1)m_2=0 $$ Find integers $t,n_1,...,n_s \in \mathbb{N_0}$ and $\lambda_1,\...
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  • 576
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Can one check if a map of infinite rank free modules is an isomorphism modulo the maximal ideal

If $(A,\mathfrak{m},k)$ is a commutative local ring, $M,N$ are two free modules, and you are given a map $M\to N$ such that the induced map $M\otimes k\to N\otimes k$ is an isomorphism, then was $M\to ...
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How can I explain the intuition behind the First Isomorphism Theorem for modules or for rings to a lay person in mathematics

How can I explain the intuition behind the First Isomorphism Theorem for modules or for rings to a lay person in mathematics, like to a non-scientist. I want to use a very elementary example in life. ...
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Distribution of a matrix product under modular arithmetic

The question I have is related to modular algebra. All algebraic statements I make henceforth are with respect to algebra of the module where arithmetic is over $\mathbb{Z}_q$ for general $q \geq 2$. ...
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Let $M = \operatorname{cok}(\phi)$ where $\phi: \mathbb{Z}^3 \to \mathbb{Z}^3$. Express $M$ as a direct sum of cyclic modules.

Let $\phi: \mathbb{Z}^3 \to \mathbb{Z}^3$ be given by the matrix $A$ below. $$A = \begin{bmatrix} 2 & 2 & 4\\ 2 & 4 & 6 \\ 2 & 6 & 8 \end{bmatrix}$$ If $M = \operatorname{cok}(\...
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Covariant functor preserves chain homotopy

Let $C_\bullet$ and $C'_\bullet$ be two chain complexes of $A$-modules, $A$ a $K$-algebra, and $f_\bullet,g_\bullet: C_\bullet \to C'_\bullet$ two homotopic morphisms. Show that, for any covariant ...
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Short Exact Sequence Involving Ext Functor

Let $A$ be a $K$-algebra, and $0 \to M' \to M \to M'' \to 0 $ be an exact sequence of $A$-modules that splits. Show that $$ 0 \to \operatorname{Ext}^n_A(X,M') \to \operatorname{Ext}^n_A(X,M) \to \...
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Is submodule of a finitely generated free $\mathbb{Z}G$-module free when $G$ is free?

We know that a submodule of a free $R$-module is not necessarily free. But if $R$ is a PID, then every submodule of a free $R$-module is free. Now let $G$ be a free group of finite rank $r>1$ and $...
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$\operatorname{Hom}(L, \operatorname{Hom}(M,N)) \cong \operatorname{Hom}(M, \operatorname{Hom}(L,N))$

Let $L, M,N$ be three $A$-modules, $A$ a commutative ring. Show that $$\operatorname{Hom}_A(L, \operatorname{Hom}_A(M,N)) \cong \operatorname{Hom}_A(M, \operatorname{Hom}_A(L,N))$$ I know that, by ...
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The proof of the Hilbert-Burch theorem

I am reading the proof of Theorem 1.4.17 (Hilbert-Burch theorem) in Cohen-Macaulay Rings by Bruns & Herzog. My question Let $R$ be a Noetherian ring, and $\varphi\colon R^n \to R^{n+1}$ an $R$-...
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Modules over the trivial ring [duplicate]

I know that a module is the ring-theoretic analogue of a vector space. My question is, is every module over the trivial ring $\{0\}$ a singleton set? Or can there be a module over the trivial ring ...
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Bimodules over a categorical monad

In category theory a monad consists in an endofunctor $M\colon \mathcal{C}\rightarrow \mathcal{C}$ together with natural transformations $\mu \colon M \circ M \Rightarrow M$ and $\eta \colon Id_{\...
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Grade and projdim of finite modules

I am reading Cohen-Macaulay Rings by Bruns & Herzog, and trying to prove the lemma below: Lemma. Let $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. Then for all $\...
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Demonstrate that $\mathbb{C}[\mathbb{Z}_{2} \times \mathbb{Z}_{3}] \simeq \mathbb{C}^{6}$.

This is a review problem that I'm solving. I have been told that $ \mathbb{C}[\mathbb{Z}_{2} \times \mathbb{Z}_{3}] $ is a $ \mathbb{C} $ algebra so a vector space where the field is $\mathbb{C}$ and ...
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Submodules of a complex vector space and invariants of an endomorphism

Let $V=\mathbb{C^n}$ be a finite dimensional vector space and $f\in End(V)$. Consider the $\mathbb{C[x]}$-module $\mathbb{C_f^n}$ which is obtained by giving $V$ the left-module structure as $p.v=p(f)(...
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Chinese Remainder Theorem in Structure Theorem for f.g. Modules over a PID

I have been asked by my Algebra professor, to explicitly determine the Chinese Remainder Theorem in the proof of the Structure Theorem for f.g. Modules over a PID. Here's what I know: $\textbf{...
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Understanding natural $R$-Algebra structure on some abelian groups.

Definition ($R$-Algebra): Let $R$ be a commutative ring with a ring homomorphism $\phi : R\to S$ such that $\phi(R)\subseteq Center(S)$, then the ring $S$ is a $R$-Algebra Base Question: $R$ is a ...
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Are "modules over an additive group" a thing?

Googling around, the only notion of "module over a group $G$" I was able to find was here, where $G$ is a multiplicative group but acts on an additive group. But I'm curious about another ...
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Is a finite ring map of finite presentation finitely presented?

Let $\phi: A \rightarrow B$ be a ring map (of commutative unitary rings). Assume that $\phi$ is finite, i. e. $B$ is finitely generated as an $A$-module, and $\phi$ is of finite presentation as in ...
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Decomposition of the ring of square matrices into indecomposable submodules [duplicate]

Let $R=K^{n\times n}$ be the ring of square matrices over field $K$, alongside left-multiplication we observe $R$ as a module over itself. Find a decomposition of $R$ as a direct sum of ...
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Exact sequence in which $M/\operatorname{Ker}f$ and $\operatorname{Im}f$ appear [closed]

Let $R$ be a ring and $M,N$ be $R$-module. Let $f:M→N$. I'm looking for an exact sequence in which $M,N,\operatorname{Im}f,M/\operatorname{Ker}f$ appear. Do you have any good ideas? Thank you for your ...
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How do I show that if $R=\Bbb{Z}/6\Bbb{Z}$ and $M=\Bbb{Z}/3\Bbb{Z}$ then $M$ is a projective $R$-module?

Projective Module Let $R$ be a ring and $M$ an $R$-module. We say that $M$ is projective if $Hom(M,-)$ is exact. Let $R=\Bbb{Z}/6\Bbb{Z}$ and take $M=\Bbb{Z}/3\Bbb{Z}$. Using the above definition I ...
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sets and logic question

Let S = {1, 2, 3, …, 19, 20}. Let ≡ be the equivalence relation on S defined by congruence modulo 7. a) Find the quotient set 푆 ≡. b) Find a system of equivalence class ...
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1 answer
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If $M \otimes N$ is flat and $M$ is flat, then $N$ is flat?

So I was wondering if the product of two modules is flat then one of them being flat implies the other one is flat too. I think the answer is affirmative and I will try to present my reasoning here. I ...
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Definition of 'Group action commutes with endomorphism'

Let $G$ be a group and $G$ acts on $R$-module $M$($R$ is a ring). And $M$ has endomorphism $f$. In these settings, what is the definition of $f$ commutes with $G$-action ? I often hear this kind of ...
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Motivation for looking at the product module

Let $M$ be an $A-$module, $A$ a commutative ring with identity. Then we have the following result that , $\frac{A}{I} \otimes M \simeq \frac{M}{IM}$ for any ideal $I$ in $A$. This simply follows by ...
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Is the Ext module $\mathrm{Ext}^i_R(R/I^n,-)$ annihilated by a power of $I$?

Let $R$ be a Noetherian ring and $I$ an ideal in $R$. Let $M$ be a finitely generated $R$-module. Is, for all $i$ and $n$, the module $\mathrm{Ext}^i_R(R/I^n,M)$ killed by a power of $I$? I've read ...
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Tensor Algebra of Direct Sum

If $A$ is a $k$-algebra, and $(T(M), j_M)$ denotes the tensor algebra of an $A$-bimodule $M$, $j_M: M \to T(M)$ the associated homomorphism of $A$-bimodules, how can I prove that $$T(M \oplus N)\cong ...
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Elements of $\mathrm{Hom}_{\mathbb{Z}}(\prod_{i\geq 0}\mathbb{Z}, \mathbb{Z})$ vanish on almost all elements of "standard basis" $\mathbb{e}_{n}$

I've been struggling with the following exerciese: Let $f\in\mathrm{Hom}_{\mathbb{Z}}(\prod_{i\geq 0}\mathbb{Z}, > \mathbb{Z})$, where $\prod_{i\geq 0}\mathbb{Z}$ denotes the $\mathbb{Z}$-module ...
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Could you give me an example of $R$module $M$ which is finitely generated but not finitely presented? [duplicate]

It is well known that finitely gpresented module is finitely generated. But in general I heard finitely finitely module is not always presented represented. Here, finitely represented $R$ module $M$ ...
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An example of $R$module $M$ which is finitely presented but not finitely generated [duplicate]

It is well known that finitely generated module is finitely presented. But in general I heard finitely represented module is not always finitely represented. Here, finitely represented $R$ module $M$ ...
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1 answer
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Proposition 2.18 in Atiyah & MacDonald's Comm. Algebra

I understand the proof of the proposition, but I don't see how the functions in the tensored sequence will end up being $f\otimes 1, g\otimes 1$. In the first step of the proof, the functions change ...
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Alternative proof for a sufficient condition for $x$ to be integral over a ring $A$.

I have a request for an alternative proof of a fact in ring theory. Given an extension of rings $R \supseteq A$, we say that $x\in R$ is integral over $A$ if there is a monic polynomial $p$ with ...
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1 vote
1 answer
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Tensor product of modules over Kronecker algebra

Let $\mathbf{k}$ be a field and $A=\begin{bmatrix}\mathbf{k}&0\\\mathbf{k}^2&\mathbf{k}\end{bmatrix}$ be the Kronecker algebra. Let $M$ and $N$ be the left and right $A$-modules (respectively),...
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2 answers
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How do I check if a module $M=\Bbb{Z}/6\Bbb{Z}$ is finitely presented?

Let $R=\Bbb{Z}$ and take $M=\Bbb{Z}/6\Bbb{Z}\in Mod_R$. I want to check if it is finitely presented. I have the following definition: If $M$ is an $R$ module generated by $(x_i)_{i\in I}$ then we ...
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2 votes
0 answers
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Homology of the diagonal sequence of 3x3 commutative diagram of modules

Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules), and suppose that we have a 3x3 commutative diagram, where ...
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3 votes
1 answer
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An example of abelian ring which is NOT square-free.

Recall that a ring $R$ is abelian if all idempotents are central. Recall that a right module $M_R$ is called square-free if whenever $A$ and $B$ are submodules of $M$ with $A\cong B$ and $A\cap B=0$ ...
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5 votes
1 answer
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Why do we care about flat and projective modules?

I was reading the definitions of projective modules and flat modules and found myself a bit unenlightened (by all of their equivalent definitions). At least the Wikipedia articles for these classes of ...
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1 vote
1 answer
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Connection between kernels of linear maps of semimodules and injectivity

Let $S$ be a semiring (i.e. satisfies all the ring axioms besides existence of additive inverses) and $M, N$ semimodules over $S$ (same thing). For a linear map $\varphi : M \rightarrow N$, we can ...
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  • 402
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1 answer
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Do we always have linear right inverses of surjective linear functions in non-free modules?

Let $R$ be any ring, $M$ an $R$-Module, not necessarily free. Let $\varphi : M \rightarrow M$ be linear and surjective. By surjectivity, there exists a function $\varphi^*$ such that $\varphi \varphi^*...
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  • 402
2 votes
2 answers
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Meaning of 'set of well-ordered sequences'

I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968. On the second page, a module $F$ is constructed,...
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2 votes
2 answers
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Is the dual of a Noetherian module Noetherian?

Say $M$ is a Noetherian module over $R$. I wanna see how far I can get without the axiom of choice, so let me clarify that I mean "satisfies the Ascending Chain Condition" by Noetherian. ...
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  • 402
3 votes
0 answers
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Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and for every $[a]_{n}$ we get $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ .

Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and every $[a] \in \mathbb{Z}_{(m,n)}$ can be regarded as an element $[a]_{n}$ such that $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ . ...
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What are conditions on a ring $R$ for all finitely generated submodules of $R^n$ with torsion-free quotients to be free?

Let $R$ be a commutative integral ring, and let $M$ be a finitely generated sub-module of $R^n$ such that $R^n / M$ is torsion-free (that is, if $rx \in M$ then $x \in M$ for all $x \in R^n$ and $r \...
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1 vote
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Admissibility necessary for a $(\mathfrak{g}, \mathfrak{k})$-module to be a direct sum of simple $\mathfrak{k}$-modules?

I am now looking over the book A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Volume 67 of Mathematical Surveys and Monographs, AMS. I would ...
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  • 370
1 vote
1 answer
38 views

Is every quotient module with identity cyclic?

If $R/\langle a \rangle$ is the quotient module over the commutative ring $R$ with identity 1, then, $R/\langle a \rangle=\{r+\langle a \rangle: r \in R\}=\langle1+\langle a \rangle \rangle$ Is it ...
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