Mathematics Stack Exchange
Mathematics Stack Exchange

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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Group action by symmetric group $S_k$ on $k$-tensors $\mathcal{T}^k(M)$ for an $R$-module $M$ over a unital commutative ring $R$.

I was puzzled at first, over the fact that we define the group-action of the symmetric group $S_k$ on $k$-tensors $$z \in \mathcal{T}^k(M)$$ as $$\sigma z = \sigma(m_1 \otimes \cdots \otimes m_k) = (m_{\sigma^{-1}(1)} \otimes \cdots \otimes…
Ben123
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Is there always a free $\mathbb{Q}$-module on a set $A$?

As the title says; Is there always a free $\mathbb{Q}$-module on a set $A$? I believe so, since you can always form the free $\mathbb{Z}$-module on a set $A$, $F(A)$ and then identify $\mathbb{Z}$ as a subset of $\mathbb{Q}$.
Ben123
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Modules with no infinite direct sums of simple summands.

Assume that $M$ is any right $R$-module. Assume that $M$ has no infinite direct sums of simple summands. Does this assumption imply that $M$ has no infinite set of pairwise isomorphic simple summands ?!. Here is my attempt: Assume on the contrary…
Hussein Eid
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If a module $M$ has DCC on essential submodules, then $\mathrm{soc}(M)$ is an essential submodule of $M$.

Let $R$ be a ring with $1$ and $M_R$ any right $R$-module. Assume that $M$ has the descending chain condition on essential submodules. Let $\mathrm{soc}(M)= \bigcap_{i\in I}A_i$ be the intersection of all essential submodules of $M$. How can I prove…
Hussein Eid
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An endomorphism ring acting on a module from the right

First of all, excuse me if the question seems stupid. But I want to understand what is written in this screenshot. The book name is (Extensions of Rings and Modules by Birkenmeier, Park, and Rizvi). The authors say that $Q$ is acting on $E$ from the…
Hussein Eid
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Do pseudo-isomorphisms of $R$-modules satisfy the 2 of 3 property?

Suppose we have $L \xrightarrow{f} M \xrightarrow{g} N$ where $L,M,N$ are $R$-modules and $f,g$ are homomorphisms. A pseudo-isomorphism of $R$-modules is a homomorphism with finite kernel and cokernel. Is it the case that if 2 out of the 3 maps…
Austin Maison
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Direct consequence of a Lemma used in definition of Quotient Module

I have the following Lemma already proved. If $A$ is a submodule of the $R$-module $X$, then $\{r(x+a) | a \in A \} \subset rx+A$. Then in the same book it is said that as a direct consequence of this Lemma, the coset $rx+A$ depends only of $r \in…
Paulo Henrique L.A.
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A summand of a GCS module is a GCS module

Recall that a right $R$-module $M$ is CS if for every submodule $N\subseteq M$ there exists a summand $L \subseteq^{\oplus} M$ such that $N$ is an essential submodule of $L$. It's well-known that a summand of a CS module is again CS (see, for…
Hussein Eid
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Seeking for an example of a non-Noetherian module having a Noetherian submodule.

We know that a $R$-module $M$ is Noetherian if a submodule $N$ and its Quotient module $M/N$ is Noetherian. Now I want a counter example where a submodule is Noetherian but the original module is not. Please any help will be appreciated.
Md Saiful Islam
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$p$-primary subsmodule of $\mathbb{Z}$

Let $p^r \mathbb{Z} (r\geq 2)$ be a submodule of $\mathbb{Z}$. Is it a $p$-primary subsmodule of $\mathbb{Z}$? Please guide me with a answer. Thank you for your kindness.
Saniya
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An example of a simple module which does not occur in the regular module?

Let $K$ be a field and $A$ be a $K$-algebra. I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable $A$-modules, that is $A=\oplus_{i=1}^n S_i$ where each…
Math137
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Can there exist an element $r$ in the base ring $R$ such that $r.m =0$ for all $m$ in a module $M$ over $R$?

Let $M$ be a nontrivial $R$ module. Then can there exist an element $r \in R$ such that $r.m=0$ for all $m \in M$?
Saikat
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Modules over the trivial ring

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 that has higher cardinality than $1$?
user107952
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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 correct? If it is ,then it means that every quotient…
Esha
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Find $n \in \mathbb{N}$ which makes $x_1, x_2, ..., x_{26} \in \mathbb{N}$ exist which satisfies $\sum_{k=1}^{25} n^{x_k}=n^{x_{26}}$.

Find $n \in \mathbb{N}$ which makes $x_1, x_2, ..., x_{26} \in \mathbb{N}$ exist which satisfies $\displaystyle \sum_{k=1}^{25} n^{x_k}=n^{x_{26}}$. My attempt: \begin{align} &\sum_{k=1}^{25} n^{x_k} \equiv 25 (\text{mod } {n-1}). \\ &n^{x_{26}}…
RDK
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