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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Is every R-module a principal ideal generated by the smallest element in the module?

The question is self-explanatory. If we have a an R-module $V$, where $R$ is a ring, every element in $V$ can be represented as $r*x$ [by division algorithm], where $r$ is any element from $R$ and $x$ is the smallest element in the module. So is the…
user67803
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Examples of non semisimple module

I've seen following definition: an $R$-module $M$ is semisimple if every submodule of $M$ has a complement. Does anyone have example of a module which is not semisimple in $\mathbb{Z}$, $\mathbb{C}[t]$ and $\mathbb{C}[\mathbb{Z}]$? I think…
user709182
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Is $\hom(\bigoplus_i V_i, \bigoplus_j W_j) \cong \bigoplus_i \bigoplus_j(V_i,W_j)?$

Given the (external) direct sum of $R$-modules: $V= \bigoplus V_i, W= \bigoplus W_j$, will $\hom(V,W) \cong \bigoplus_{i,j} \hom(V_i,W_j)?$ I know that $\hom(V,\bigoplus_j W_j) \cong \bigoplus_j \hom(V,W_j)$, the isomomorphism given by $\Phi(f)…
user2345678
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If $M$ is an $R \times S$-module, then $M = M_{1} \oplus M_{2}$ where $M_{1}$ is an $R$-module and $M_{2}$ is an $S$-module

I was writing the proof of the fact that if $R$ and $S$ are semisimple, then so is $R \times S$, where I tried showing that if $M$ is any $R \times S$-module, then M is semisimple. To show this, if we show that $M$ is isomorphic to $M_{1} \oplus…
P-addict
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Nonzero module with no nonzero finitely presented submodule

My question is: Does there exist a nonzero module over a non-Noetherian ring with no nonzero finitely presented submodule? For any element $m$ of a left (right) $R$-module $M$, the submodule $Rm$ ($mR$) is finitely generated, but not finitely…
Geoffrey Trang
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Showing that the kernel of the localisation map of a module homomorphism is the localisation of the kernel?

Let $ \phi : M_1 \to M_2 $ be an $A$-module homomorphism. Prove that $\phi$ is injective iff $\forall \mathfrak{m} \in \text{Max}(A)$, $\phi_{\mathfrak{m}}$ is injective. I am aiming to show that $\text{ker}(\phi_{\mathfrak{m}}) \subset…
nuiglate
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the only $R$-module homomorphism $R/(p_1^{e_1}) \to R/(p_2^{e_2})$ is the zero homomorphism

Let $R$ be a PID. and let $p_1,p_2$ be irreducible elements with $(p_1) \neq (p_2)$. Let $e_1,e_2$ be non-negative integers. I want to show that the only $R$-module homomorphism $$ R/(p_1^{e_1}) \to R/(p_2^{e_2}) $$ is the zero homomorphism. My…
Sigurd
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Let R be a commutative ring with a 1, and let J and K be ideals of R. Prove that $_R(R/J) \cong {_R}(R/K)$ iff $J = K$.

This is a question taken from p.95 of Hartley and Hawkes: Rings, modules and Linear Algebra. Firstly is the question correct or should it read $J \cong K$? This would seem more natural to me. Assuming the question is correct then one of the…
G Aker
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Pathological examples of finitely generated modules

Let $R$ be a commutative (noetherian) ring with identity. Let $M$ be a finitely generated $R$-module. Let $n\in\mathbb{N}$. Does $$M=\bigoplus_{k=1}^n M_k,$$ where $M_k$ is an either an ideal or quotient ring of $R$. If not, are there any explicit…
Ishaan Shah
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How to find if there's exists an integer X that fullfil this equation

Given A, B, C & D and the equation : (A*X + B)%D = C I tried to move the modulo to the other side but I end up with 2 unknown variables.
Andrean Lay
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Question about $\text{Hom}_{\mathfrak{g}}(M,L)$

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Suppose $M,N,L$ are $\mathfrak{g}$-modules, $N$ is a $\mathfrak{g}$-submodule of $M$. Does this implies $\text{Hom}_{\mathfrak{g}}(N,L)\le \text{Hom}_{\mathfrak{g}}(M,L)$ as a vector…
James Cheung
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Find submodules with dimension equal 2.

I have the following problem Consider the vector space $\mathbb C^3$, the matrix $$A=\begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 &0 \end{pmatrix}$$ $\mathbb C^3$ is made into a $\mathbb C[T]$-module where $T$ acts like $A$. Denote by $L(v)$ the…
Alejandro Dz
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Condition on product implies that is it the trivial module?

Let $M$ be an $A$-module and let $\mathfrak{a}$ be an ideal of $A$. Suppose that $\mathfrak{m} \cdot M =0$ for every maximal ideal $\mathfrak{m}$ of $A$ such that $\mathfrak{a} \subseteq \mathfrak{m}$. Is $M$ the trivial module? Here $\mathfrak{m}…
user6495
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Dual and simple modules

Let $A$ be a $k$ finite dimensional algebra and let $M$ be a simple finite dimensional right $A$-module. Why is the dual of $M$, i.e $\operatorname{Hom}_{k}(M,k)$ a simple left $A$-module?
user10
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A property of the reject of modules

Let $M$ and $X$ be two finitely generated modules over a finite dimensional algebra $A$. The reject of $M$ in $X$ is $$rej_M(X)=\cap_{f:X \rightarrow M} Ker f.$$ I have seen in a place that if $Hom_A(X,M) \not =0$, then there is a monomorphism…
Xiaosong Peng
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