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Questions tagged [modules]

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

Modules are abelian groups with an added notion of multiplication by elements in a ring. They generalize abelian groups, which are modules over the integers, and vector spaces, which are modules over a field.

Rigorously, a left $R$-module is defined as an abelian group $M$ paired with a ring $R$ with a binary operation from $\cdot\;\colon R\times M\rightarrow M$ satisfying the following axioms for all $m,n\in M$ and $r,s\in R$:

  1. $r\cdot(m+n)=r\cdot m+r\cdot n$

  2. $(r+s)\cdot m=r\cdot m+s\cdot m$

  3. $(rs)\cdot m=r\cdot(s\cdot m)$

If $R$ is a unital ring, we often also require that $1\cdot m=m$.

A right module is defined similarly by rewriting the axioms with the ring elements acting on the right side.

Modules often arise in the study of commutative rings and in algebraic geometry, but may appear in any investigation of the structure of a ring as a result of the Yoneda embedding which sends a ring to the category of left modules over that ring.

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For an integral domain $R$, is $S^{-1}(R[x])$ necessarily not a projective $R[x]$ module?

First, let $K$ be a field. Then the field of rational functions $K(x)$ is a projective $K[x]$ module. I can prove this by showing that $K(x)$ is not a free $K[x]$-module because $K[x]$ is a principal ideal domain, which means that free implies…
Poko
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Isomorphism of R-modules, (X), (Y), (X,Y)

Let $K$ be a field. Let $R=K[X,Y]$. Observe the ideals $(X), (Y), (X,Y)$ as $R$-modules. Which of them are isomorphic, which are not? My guess is, that $(X)\cong (Y)$ and $(X)\ncong (X,Y)$, $(Y)\ncong (X,Y)$. So I want to give a homomorphism of…
Cornman
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Examples for a K-module with K ring, but not an integral domain?

I am looking for an example for such a module, because I want to find a module for which the set of torsion elements is not a submodule of $M$.
MPB94
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show that the torsion is a submodule

Let $R$ a unital ring (not necessarily commutative) and $M$ a $R-$module. Let $$Tor(M)=\{m\in M\mid \exists r\in R, r\neq 0: rm=0\}.$$ 1) Show that if $R$ is a domain then $Tor(M)$ is a $R-$submodule. 2) Show that if $R$ is not a domain, $Tor(M)$ is…
user386627
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Simple question about R-modules

We have been given R, a semi-simple ring. My notes state without any justification that every R-module M is a quotient of a direct sum of copies of R. I guess this is supposed to be obvious but I can't see why this is. Thanks for any help figuring…
Pierre
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The simple modules of upper triangular matrices.

Let R be the ring of $n\times $n upper triangular matrices over a field Q ,and let $\wp$ be the class of simple left R-modules.Prove that: $(1) Tr_{R}(\wp)=\{[[a_{ij}]]\in R\mid a_{ij}=0 (i\geq2)\}.\\$ $(2) Rej_{R}(\wp)=\{[[a_{ij}]]\in R\mid…
guojm
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Show every non-zero submodule of $M$ contains a minimal submodule iff SocM is essential in M.

I am studing the book "Rings and Categories of modules" written by Frank W. Anderson and Kent R. Fuller. On page 119, I am at a loss for the Corollary 9.10. Corollary 9.10. Let $M$ be a left R-module. Then SocM is essential in M iff every non-zero…
Daisy
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On the module over cyclotomic fields.

Let $\zeta$ be the $p^l$-th root of unity in the complex plane, where $p$ is a prime number. Suppose $M$ be a finitely generated $\mathbb{Q}(\zeta)$-module. Is it true that $\dim_\mathbb{Q} M\geq \frac{1}{p^l} \dim_{\mathbb{Q}(\zeta)} M$?
user9384023
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Direct summands of $\mathbb Z \oplus \mathbb Z$

Consider the $\mathbb Z$-module $\mathbb Z \oplus \mathbb Z$. I have seen two assertions about the direct summands of $\mathbb Z \oplus \mathbb Z$ but I have trouble with these: $(a,b)\in\mathbb Z \oplus \mathbb Z$ spans a direct summand if and…
Bobby
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$R=F[x]$, $A=[v_1,\ldots,v_n]\in \operatorname{Mat}_n(R)$, $M=R^n/(Rv_1+\cdots+Rv_n)$ Then $M$ is finite dimensional over $F$ iff $f=\det(A)\ne 0$

Let $R=F[x]$ where $F$ is a field, the $n\times n$ matrix $A=[v_1,\ldots,v_n]\in \operatorname{Mat}_n(R)$. Let $M=R^n/(Rv_1+\cdots+Rv_n)$ be the $R$-module. Show that $M$ is finite dimensional over $F$ iff $f=\det(A)\ne 0$, and in this case,…
Edelweiss Ntu
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Why $0\rightarrow B\otimes I\rightarrow B\otimes R$ is exact?

Let $B$ is a right $R$-module. Show that $$0\rightarrow B\otimes I\rightarrow B\otimes R$$ is exact for every finitely generated left ideal $I$. Any help is appreciated, thanks a lot.
unicornki
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Definition of $\oplus_{i\in\mathbb I}R$ when $R$ is a module.

Let $\{R_i\}_{i\in I}$ a collection of $M$ modules. I know that if $\{e_1,...,e_m\}$ is a basis of $\oplus_{i=1}^nR_i$, then, $$x\in \oplus_{i=1}^nR_i\iff \exists ! a_1,...,a_m\in M: x=a_1e_1+...+a_me_m.$$ Now, I have a definition when $I$ is…
MSE
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Proposition 19.6 of Lam about cogenerators of $\mbox{Mod}_R$

I have a question about proposition 19.6 of Lam in his book "Lectures on Modules and Rings (1999)", on page 507-508. The proposition is about cogenerators of $\mbox{Mod}_R$. The proposition is as follows: For $U_R\in \mbox{Mod}_R$, the following are…
Steenis
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Direct Sum and Sum

We know if $R$ is semi simple ring and $M$ is $R$-module then for any submoldule $N$ of $M$ there exists submodule $N'$ such that $N \oplus N'=M$. Now I have trouble with sum and its supposed answer. let $R$ be ring (Not semi simple) and $M$ is…
Bobby
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Why is $0_R0_M$ = $0_M$ in a left $R$-module $M$?

Given a left module $M$ over a ring $R$ where $0_M$ is the additive identity in $M$ and $0_R$ is the additive identity in $R$, I am trying to work out why $r0_M= 0_M$ for all $r \in R$. So far I have \begin{align*} & r0_M = r(0_M+0_M)\\ …
Anfänger
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