Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
10,107 questions
0 votes
1 answer
55 views
A module is artinian and noetherian iff it has finite length
As stated in the title, I am trying to prove that (I am assuming $R$ is a commutative ring with unity); Let $M$ be an $R$-module. Then $M$ is artinian and noetherian if and only if $\ell \left( M\...
- 195
-1 votes
0 answers
29 views
Exact sequence question about direct sums and kernels and images [closed]
$M_i$ and $M_i'$ s are $R$-modules. The arrows are $R$-homomorphisms. The diagrams are comutative. $M_r \longrightarrow M \longrightarrow M_r'$ are exact sequences such that $i_r:M_r \longrightarrow M$...
- 29
2 votes
1 answer
71 views
Proving equal length of a composition series constructed via submodule intersections and sums with a given composition series
This is a followup question to this question I asked yesterday (which by the way I was almost able to figure out entirely), in which I tried to prove that: Let $R$ be a commutative ring with identity ...
- 195
0 votes
0 answers
58 views
Invariance of the length and the factors (up to isomorphism) in the composition series of a module
I aim to break down the proof of the Jordan–Hölder theorem in the context of finitely generated $R$-modules. Let $R$ be a commutative ring with $1$, and let $M$ be an $R$-module. In the lecture, we ...
- 195
2 votes
0 answers
95 views
Is the natural isomorphism $\mathrm{Hom}_R(\bigoplus A_i, \prod B_j) \cong \prod_{i,j} \mathrm{Hom}_R(A_i, B_j)$ an isomorphism of modules?
It is a fundamental result in module theory that for any families of $R$-modules $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$, there is a natural isomorphism of abelian groups: $$ \mathrm{Hom}_{R}\left(...
- 105
3 votes
1 answer
124 views
Adjointness of Hom and Tensor: understanding statement
A statement made in the class was as follows: Theorem Let $A,B$ be rings (may be non-commutative). Let $X_B$ be right $B$-module. Let ${}_BY_A$ be $(B,A)$-bimodule. Let $Z_A$ be right $A$-module. ...
- 3,485
1 vote
1 answer
83 views
$M$ is a $\mathbb{F} [x]$ module iff $M$ is a $\mathbb{F}$ vector space with a linear map $\varphi : M \to M$ s.t. $f\ast v = [f(\varphi)](v)$
I aim to prove the following claim: Claim - Let $\mathbb{F}$ be a field and let $R = \mathbb{F} \left[x \right]$ be the polynomial ring over $\mathbb{F}$. Then $M$ is a $R$-module if and only if $M= ...
- 889
1 vote
1 answer
70 views
Associated ideals of noetherian rings
I want to prove the following statement: Let $A$ be a noetherian ring. A prime $\mathfrak{p}$ of $A$ is an associated prime of $A$ if and only if there is some element $f \in A$ with $\mathrm{Supp}(f) ...
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