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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If $A/B \cong C/B$, then is it true that $A\cong C$?

Let $A, B$ and $C$ be $R$- submodules of an $R$- module $M$ and $A/B \cong C/B$. Then is it true that $A\cong C$? Here $B$ is a submodule of both $A$ and $C$.
uuuuuuuuuu
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number of possible module structures on this abelian group.

So I am being asked how many $\mathbb{Z}[x]$ module structures does the abelian group $\mathbb{Z}/5\mathbb{Z}$ have. Thinking about it, we really only need to define the action of $x$ on $[1]$ because if we do so, then we know what the action of…
Daniel Montealegre
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Is Invariant Basis Number a simmetric condition?

A ring $R$ has Invariant Basis Number if, for every $n,m \ge 0$, $R^n \cong R^m$ (as right R-modules) implies $n=m$. I want to show that that R has Invariant Basis Number if and only if $R^n \cong R^m$ (as left R-modules) implies $n=m$. Is it true…
effezeta
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What is a good way to prove a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule?

The take-home exam give me the following problem: Prove that a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule. For the if side, I assume $M$ is finitely generated and we have a…
Bombyx mori
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What information about a ring does the category of modules of this given ring contain?

I apologize in advance for my vague question. For an algebra course I am taking at the moment we have to write a short paper about a chosen subject, I chose to prove the Morita equivalence of finite dimensional $K$-algebra with a bound quiver…
M.v.Roozendaal
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Is $I=(x,y)$ seen as an $\mathbb C[x,y]-$module is free

Is $I=(x,y)$ seen as an $\mathbb C[x,y]-$module is free ? I would say that $\{x,y\}$ is a basis but since $x$ and $y$ are in $\mathbb C[x,y]$, may be it's not true.
user386627
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A module is both left artinian and left noetherian,but it is neither right artinian nor noetherian,

I am reading Rings and Categories of Modules by Frank W.Aderson,on 130 pages. It ssys,the ring R of all 2$\times$2 upper triangular matrices \begin{Bmatrix} a & b \\ 0 & \gamma \end{Bmatrix} with a,b$\in$R and $\gamma \in$Q is both left artinian…
guojm
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Determining some properties of $\mathbb{C}^3$ as a $\mathbb{C}[x]$ module

I'm wondering how I could answer the following questions about $N = \mathbb{C}^3$ with a $\mathbb{C}[x]$-module structure given by the linear map $A$ on $\mathbb{C}^3$: $$A = \left(\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 &…
pizzaroll
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Irredundant family of generators not a basis

I would appreciate help (self-studier) with an example in Ried's "Undergrad. Commutative Algebra" p.41 in the chapter on modules. "Even when $M$ is free, an irredundant family of generators is not necessarily a basis; for example $A$ = $k[X]$, $M =…
user12802
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Injective integral domains are fields

Prove that if an integral domain $R$ is an injective $R$-module then $R$ is a field. "Proof" Choose a non-zero element $x$ of $R$ and consider the map $f: R \rightarrow R$ given by $f(t)=tx$. This yields an exact sequence $0 \rightarrow R…
user10
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How can I prove that $\mathbb Z[x]/(1+x^2)\mathbb Z[x]$ is a free module with basis $\{1,\bar x\}$?

1) How can I prove that $\mathbb Z[x]/(1+x^2)\mathbb Z[x]$ is a free $\mathbb{Z}$-module with basis $\{1,\bar x\}$? I wanted to prove that $$\mathbb Z[x]/(1+x^2)\mathbb Z[x]\cong \mathbb Z^2,$$ but it looks complicate. 2) Is $\mathbb Z[x]$ a free…
user386627
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How to show flat modules stay flat after base ring extension?

If $g: A\rightarrow B$ is a ring homomorphism and $M$ is a flat $A$-module. Then $M_{B}=B\otimes_{A}M$ is flat $B$-module. We need to prove if $f:S\rightarrow T$ is injective, then $f\otimes 1: S_{B}\otimes( B\otimes_{A}M)\rightarrow T_{B}\otimes…
Bombyx mori
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Sub-modules of finitely generated free $\mathbb{Z}$-module

Suppose $M$ is a finitely generated free $\mathbb{Z}$-module of rank $n$, so $M=u_1\mathbb{Z} \oplus \cdots \oplus u_n\mathbb{Z}$. Suppose that $M'$ is a submodule of full rank, so it is also rank $n$, and let $\widetilde{M}=u_1\mathbb{Z} \oplus…
G Tony Jacobs
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Hom and direct sums 3

Let $\{M_i\}_{i \in I}$ and N be left R modules where R is not necessarily commutative. Then how can we prove that $Hom_R(N, \bigoplus_{i \in I} M_i)$ is isomorphic to $\bigoplus_{i \in I}Hom_R(N,M_i)$. If I start from $f \in Hom_R(N, \bigoplus_{i…
budi
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Example needed for $\alpha M=M$, where $M$ is a module, and $\alpha$ an ideal of the ring

Let $\alpha\lhd R$ be a non-trivial ideal of ring $R$, and let $M$ be an $R$-module. Could someone give me an example where $\alpha M=M$? Motivation: Nakayama's lemma has this as an assumption.
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