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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Can we find a module homomorphism satisfying certain conditions?

Let $M$ and $N$ be $R$- modules and $M'$ be a submodule of $M$. Let $v\in M $ but $v\notin M'$. My question is -- Can we find a module homomorphism $\phi$ from $M$ to $N$ such that $$\phi(M')=\{0\}$$ but $$\phi(v)\neq0.$$ We already know that this…
uuuuuuuuuu
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Proving canonical isomophism of homs

I have following problem: let $R$, $A$ be rings, $M$ a R-Module. I have to prove that $\tau: \operatorname{Hom}_{Ab}(M,A) \to \operatorname{Hom}_{R-Mod}(M,\operatorname{Hom}_{Ab}(M,A)) $ is an bijection for $\tau$ which is defined as $\tau(f)(m) =…
user267839
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Rank of a module the Structure Theorem and Smith Normal Form

Theorem 211 (Structure Theorem for Finitely Generated Modules) Let M be a finitely generated R-module. Then there exists a non-negative integer r, called the (torsion-free) rank of M and non-zero, non-unit elements $d_i$ $∈$ R, known as the…
asdf
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A moduleM is finitely cogenerrated iff every module that cogenerates M finitely cogenerates M?

I am reading Rings and Categories of Modules by Frank W.Aderson,on 124 pages.There is a corollary: 10.3.corollary.If M is finitely cogenerated, then every module that cogenerates M finitely cogennerates M. there is also a example . The example…
guojm
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A question about module

Assume R is commutative. Show that an R-module M is irreducible if and only if M is isomorphic (as an R-Module) to R/I where I is a maximal ideal of R. If I remove the hypothesis that R is communtative, is it result still true?
Yuan
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a submoudule can be both essential and superfluous

enter image description here I am reading Rings and Categories of Modules by Frank W.Aderson,on 73 pages,I can't understand the statement in the picture.I can't found a submodule is both essential and superfluous.I hope someone can help me,thanks!
guojm
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Possible way to base change to a module?

Let $M$ be an $R$-module (all rings are commutative). Let $S$ be a subring of $R$. Is there anyway to figure out what are all the possible $S$-module $N$ such that $N\otimes_{S}R\cong M$? In particular, what happened if $R^{\times}\bigcap…
caveat
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Example of an $R$-module $M = M_1\oplus M_2$ and a set $S$ such that $\operatorname{span}_RS = M$ but $S\cap M_1 = S\cap M_2 = \emptyset$

Give an example of an $R$-module $M$ such that $M=M_1 \oplus M_2$, $M=\operatorname{span}_R(S)$ and $S \cap M_1 = S \cap M_2 = \emptyset$. I'm not sure if this is a very well known fact, but I can't see why $\mathbb{Z}_6$ works here. If you take…
Lotte
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Confused in Module theory

I have tried to solve an exercise regarding integral ring extensions which asked me to prove this: If each of $\{R_i\mid i=1\ldots n\}$ is a family of integral ring extensions of a ring $R$, then $\prod_{i=1}^n R_i$ is an integral extension of $R$…
Greg
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$A\times B\cong M$

Let $M$ be left $R$-module and $A$ ,$B$ are two submodule of $M$ such that $A\times B\cong M$. Is there submodule $C$ such that $A+C=M$ and $A\cap C=0$? All my attempts proving the $A$ has a complement in $M$ but it seems to be wrong, but can not…
Bobby
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How to solve $x^2 - 1$ in ring $\mathbb{Z}/700\mathbb{Z}$

How to solve $x^2 - 1$ in ring $\mathbb{Z}/700\mathbb{Z}$ I know China's theorem about mods, but i doesn't understand how to use it, when i need to find root for this problem. But i can find for example $7^{22^{34}}$ mod 567, or something like this
Eugene Korotkov
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modules is direct sum of simple submodule all of which are isomorphic to simple module

I'm looking for an article or source that deals with modules with this property: $M$ is $R$-module which is direct sum of simple submodules all of which are isomorphic to simple module. I want to know more about this module and their properties.
Bobby
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Indecomposable module, Algebra

Let R be $K[x]$, the the ring of polynomials over K, and $A \in M_n(K)$. Then $K^2$ is a R-modul by using the Matrix A, with $p(x) \circ v=p(A)v$, where $v \in K^2$ and $p \in K[x]$. Now I got show that $K^2$ for $A= \begin{pmatrix}0 & 1 \\ 0& 0\\…
Thesinus
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Uniqueness submodule of cyclic module

I'm having some problems with an exercise from Hungerford's Book of Algebra. It states: Let $R$ be a PID, and $A$ a unitary $R$-module such that $A$ is cyclic of order $r$. a) Prove that every submodule of $A$ is cyclic of order dividing $r$. b)…
Luis Ferroni
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Modules and epimorphisms

Let M be a finitely generated R-module ,where R is a PID and N a submodule.Is there an epimorphism from M to N?
t.k
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