# Questions tagged [modules]

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

6,281 questions
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### Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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### Question about associated primes and annihilator

I am trying to solve the following exercise: Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets ...
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### Prove that the following conditions are equivalent for a $_RP$ projective module

Let $_RP$ be a projective module, then: End($_RP$) is semiperfect P is semiperfect and finitely generated are quivalent. I have to prove this, but I think I'm not understanding the idea behind. ...
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### On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits

Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor $T: R$-...
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### Does taking the radical of modules commute with taking quotients?

I am studying a proof which shows that a particular $R$-module map $\pi$ is surjective onto a module $M$. The details of the map are complex, so I won't give them here, and will just sketch what I ...
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### $(\mathfrak{g},K)$-Module where $K$ is a field?

I am reading through Finite groups of Lie type_ conjugacy classes and complex characters by Roger Carter, and came across this passage where Carter is setting up a special class of module to give a ...
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### The class of left serial rings is closed under extensions

A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. ...
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### Is a finitely generated torsion-free R-module free over R if R is an integral domain?

I know this is the case if $R$ is a PID, but PID's are special instances of Integral Domains, so I am wondering if there is a counter-example to the case where R is an integral domain. This post ...
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### Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
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### Local behavior of sheaf of ideals given by a closed immersion

I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals $I_Y(U) =$ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$} is quasi coherent. I sort of understand ...
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### A problem about checking isomorphism of R-module

Let $R=K\left[x_1,x_2,\cdots,x_n\right]$ be a polynomial ring with coefficients in the field $K$; $\alpha_1,\alpha_2,\cdots,\alpha_p,\beta_1,\beta_2,\cdots,\beta_q\in R^{1\times m}$; $M_1$ be the $R$-...
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### Example of both finitely and infinitely generated free modules which are direct sum of two non free modules

Does there exist non free $R$-modules $F_0,F_1$ such that $F=F_0\oplus F_1$ be a free $R$-module? 1- If yes then for what kind of rings $R$ there exist such $R$ modules? 2- If yes then does it holds ...
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### Determinant of a nonfree module

Is there a definition of a determinant which can be applied to a module with no basis? We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. ...
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### Condition for a finitely generated flat module be projective

Prove that: Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a ...
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### Why is $M/\operatorname{rad}(M)$ semisimple?

Let $M$ be an $A$-module for $A$ a finite dimensional algebra. Let $\operatorname{rad}(M)=\bigcap\{N\subsetneq M\ \text{maximal}\}$. Clearly, $M/N$ is simple for any maximal submodule $N$. It seems to ...
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### Isomorphic algebras of endormporphisms

Let $R$ be a simple algebra, $M$ a simple $R$-module and $N$ a simple $\mathcal{M_n}(R)$-module (always considering finite dimension). Prove that $End_R(M)$ and $End_{\mathcal{M_n}(R)}(N)$ are ...
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### Exact sequence construction

Given an $R$-module $M$ arbitrary, show it is always possible to construct an exact sequence of $R$-modules $$0\longrightarrow K \longrightarrow L \longrightarrow M \longrightarrow 0,$$ with $L$ a ...
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### Cardinality of an intersection of two submodules.

Assume $p$ is a prime number and $q = p^2$. Denote by $A$ the ring $\mathbb{Z} / q \mathbb{Z}$. Consider a finite type module $M$ over $A$ whith cardinality $q^N$ where $N$ is an integer, $N>0$. ...
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### Showing that M is a finitely generated R-module

Lemma: Let $A$ be an $R$-Algebra such that $A$ is a finitely generated $R$-Module. Let $M$ be a finitely generated $A$-module. Show that $M$ is a finitely generated $R$-Module. Proof: So things I ...
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### Ideals of Dedekind rings are projective

Let $R$ be a Dedekind domain and $I$ be an ideal of $R$. Show that $I$ is a projective $R$-module. My definition of a projective module is that it is a direct summand of a free module, i.e. there ...
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### Are the two scalar multiplication on an $R$ module equivalent?

Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then, consider $Hom_{\mathbb{Z}}(R, M)$ (note that the homomorphisms are as $\mathbb{Z}$ modules). This is an $R$ module in two ...
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### Isomorphic $\mathbb{C}[X]$-modules [duplicate]

My question is : It is true that $\mathbb{C}[X]/(x-c)$ and $\mathbb{C}[X]/(x-d)$ are isomorphic $\mathbb{C}[X]$-modules if and only if $c=d$? I have the feeling that the answer is simple but I ...
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### If $I\subseteq R$ is a uniserial ideal of $R$ with finite length and $R/I$ is uniserial of finite length, then $R$ is uniserial of finite length

A module is called uniserial if the lattice of its submodules is a chain, i.e., the set of all its submodules is linearly ordered by inclusion. A ring is called right (resp. left) uniserial if it is ...