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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why is torsion free needed

This is from P134 of Rotman's Homological Algebra book. If R is a PID, then every torsion-free R-module is flat. Proof. If R is a PID, then every finitely generated R-module M is a direct sum of cyclic modules. If M is torsion-free, then it is a…
scsnm
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Tensoring is exact with free module

Let $0\to M_1\xrightarrow{f} M_2\to M_3\to 0$ be an exact sequence of $R$-modules. Show that if $N$ is a free $R$-module, then $-\otimes_R N$ is exact. This amounts to proving that $f\otimes id:M_1\otimes_R N\to M_2\otimes_R N$ is injective. Let…
Dr. Heinz Doofenshmirtz
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for $(a,b)=1$, $a^n$ to $a^{n+b-1}$ modulo $b$ is a complete system of residues of $b$ (except for the remainder 0) Is this right?

I had a discovery which, For $(a,b)=1$, $a^n$ to $a^{n+b-1}$ modulo $b$ is a complete system of residues of $b$ (except for the remainder 0) Such as, when $(a,b)=(6,17)$, the remainders are $\{6,2,12,4,7,8,14,16,11,15,5,13,10,9,3,1,6,\ldots\}$. Such…
Iris
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Let R be a ring with identity; then there is a ring isomorphism $\operatorname{Hom}_R(R,R) \cong R^{\mkern 1mu\text{op}}$.

In particular, if R is commutative, then show there exists a ring isomorphism $\operatorname{Hom}_R(R,R) \cong R$.
yajay
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Application of Module theory

Can anyone tell me the application of module . I know they are used in representation theory . What are there other places where the modules are used ???
Om Prakash
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Quotient Noetherian module

Let $M$ be a module over the commutative ring $R$ and suppose that $N$ and $H$ are submodules of $M$ such that $M/N$ and $M/H$ are both Noetherian. Show that $M/(N \cap H)$ is Noetherian. I took any arbitrary submodules $K/(N \cap H)$, I must…
Pradip
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Finding generators for a module

Suppose, if $p$ is a prime natural number, that $R$ is the set of rational numbers such that $p$ is not a factor of the denominator after cancelling out common prime factors of numerator and denominator. Consider $\mathbb{Q}$ as an $R$-module. I…
user643717
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Understanding the image of a map between two Hom sets.

I am working on a homework problem for a class and found that the part I am stuck on can be reduced to the following: Let $u: M \to M'$ and $v: N \to N'$ be module homomorphisms. I am trying to understand the map $(u,v): \text{Hom}(M, N) \to…
Yunus Syed
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problem in groups under a binary operation modular n

show that if a subset of $\big\{{1,2,...,21}\big\}$ contains an even number or contains the number $11$, then it cannot form a group under multiplication modulo $22$. first of all it's clear that the multiplication of the two numbers $2$ and $11$ is…
user689538
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C-modules via a map

Let A be the direct product ring C×C. Let τ1 denote the identity map on C and let τ2 denote complex conjugation. For any pair p, q ∈ {1, 2} define fp,q : C → C × C by fp,q(z) = (τp(z), τq(z)). Prove that if fp,q≠fp',q' then the identity map on A is…
Vikas Sharma
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if $A$ is compact $\Bbb{Z}_p$ - module such that $A/pA$ is finite, then $A$ is finitely generated over $\Bbb{Z}_p$.

Prove that if $A$ is compact $\Bbb{Z}_p$ - module such that $A/pA$ is finite, then $A$ is finitely generated over $\Bbb{Z}_p$. I have to show that every element $ x \in A$ can be written in form $x= a_1p_1 + \dots + a_np_n, \ a_1,\dots,a_n \in A, \…
Hikicianka
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Prove that two follow conditions are equivalent.

Problem: Let $M$ be a $R-$module and $N$ is a submodule of $M$ satisfies $\textit{Soc} (M) \subset N$. Prove that two follow conditions are equivalent. For each element $x \in M, x \notin N$, there is a essential submodule $B$ of $M$ includes $N$…
Minh Nguyễn Hoàng
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Is $\{ 0 \}$ a basis of the free module $\{ 0 \}$?

I'm studying modules by reading Dummit and Foote, and I'm having a problem understanding the definition of a free module. I read this stackexchange question, but I couldn't figure it out. The textbook defines a free module as following: The…
hidenoris
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Modules over Noetherian RIngs

Let $A$ be an associative Noetherian algebra (not necessarily commutative) over $\mathbb{C}$ such that every irreducible $A$-module is an injective $A$-module. Can we conclude that every $A$-module is a finite direct sum of irreducible $A$-modules?…
Ester
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Can I Find A Map from a Module M to the kernel of a map p from M to M?

I have a module homomorphism $p:M\rightarrow M$. I would like to find another module homomorphism $\phi:M\rightarrow \ker(p)$. Finding such a thing seems to be very challenging however. Is this possible? Note also that $p^2=p$.
gradintherockies
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