# Questions tagged [modules]

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

8,707 questions
Filter by
Sorted by
Tagged with
33 views

• 1,691
1 vote
47 views

### $\operatorname{Hom}(L, \operatorname{Hom}(M,N)) \cong \operatorname{Hom}(M, \operatorname{Hom}(L,N))$

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 ...
1 vote
46 views

### 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 $R$-...
• 139
1 vote
34 views

### 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 ...
• 13.6k
34 views

• 139
65 views

### 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 ...
28 views

• 163
1 vote
31 views

### 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 ...
• 2,256
56 views

### 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 ...
• 5,328
1 vote
33 views

### 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 ...
10 views

### 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 ...
• 576
1 vote
61 views

### 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 ...
• 424
48 views

### 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 ...
• 1,235
45 views

### 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 ...
82 views

### 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 ...
40 views

### 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 ...
• 592
49 views

### 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 ...
• 774
46 views

### 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 ...
• 670
33 views

### 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 ...
1 vote
28 views

### 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 ...
• 333
24 views

### 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 $M$ ...
• 592
21 views

### 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 $M$ ...
• 592
61 views

### 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 ...
• 63
1 vote
31 views

### 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 ...
• 1,738
1 vote
32 views

### 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),...
• 588
65 views

### 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 ...
• 1,235
38 views

### 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 ...
43 views

### 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 $A\cap B=0$ ...
• 527
47 views

### 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 ...
1 vote
24 views

### 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 ...
• 402
18 views

1 vote
28 views

### 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 ...
• 370
1 vote
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 ...