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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Non-simple homogeneous components of a module.

Let $M_R$ be a right $R$-module. A homogeneous component $H$ of $M_R$ is defined to be the sum $\sum_{i\in I}B_i$ where $\lbrace B_i \rbrace_{i\in I}$ is a family of mutually isomorphic simple submodules $B_i \subseteq M$. Is it true that if the…
Hussein Eid
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I can't seem to figure out how to find the inverse of 22 mod 26

I'm trying to learn how to do 2x2 hill ciphers, but I'm stuck. Through Euclidean' s algorithm I got 0=4-2*2=4-2(22-5*4) down to -13*22+11*26. The problem is when I multiply 13 by my key [1/2 3/2], it simplifies to [13/0 13/0] and none of my…
John Cantwell
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Compete direct sum

If $R$ is a ring and $A$ and $B$ are $R$-modules, then I'm familiar with the definitions of the external and internal direct sum. But I wonder what is the definition of a complete sum. Is it different from the aforementioned types of sums…
Hussein Eid
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Homogeneous components of a module

Let $R$ be a ring with unity. Can anybody please help me giving a definition of what the homogeneous component of an $R$-module $M$ is. I searched for the definition but I never got it. I appreciate any help. Thanks in advance. The citation below is…
Hussein Eid
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If $V$ is completely reducible and $End_{A}(V)$ is a division ring then $V$ is irreducible

Since V is completely reducible we can write V as a direct sum of irreducible sub modules or a direct sum of 2 complement sub modules. Can anyone give a hint on which way to write V and how can I make use of $End_{A}(V)$ as a division ring. I think …
Napu
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Prove that exits two $R-$modules $M,N$ different $0$ such that $Hom(M,N)=0$

Suppose that $R$ is a ring and $1_R$ is its multiplicative identity. Prove that exits two $R-$modules $M,N$ different $0$ such that $Hom(M,N)=0$??
Lê Nhất Sinh
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Why doesn’t this socle of the lower triangular matrix ring contain more elements?

We know that the socle of a module is the sum of all the simple submodules. I was checking this for the following example, but it doesn't seem to line up with the definitions since there are obviously simple modules of the form $(0,1,0)^T$ whereas…
scsnm
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Injectivity of images of modules to direct limits

Suppose $A_i$ is a $R$-module, with $i\in I$, $I$ being an indexed set. We assume there is a $R$-module homomorphism $\phi_{ij}:A_i\rightarrow A_j$ for every $i,j\in I$, $i\leq j$. This homomorphism has the following…
Osteo
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Thus $\dim_k(V) = |G : H|\dim_k(W)$.

Let $G$ be a finite group. Suppose that $k$ is a splitting field for all subgroups of $G$ and that $|G|$ is invertible in $k$. Let $N$ be a normal subgroup of $G$. Let $χ ∈ \operatorname{Irr}(kG)$ and $ψ ∈ \operatorname{Irr}(kN)$ such that…
scsnm
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$S \ncong {^xS}$ for all $x \in G$ \ $N$

For any subgroup $H$ of a fnite group $G$ and any $x ∈ G$, the right $kH$-module $k[xH]$ is also a left $k^{x}H$-module because $xH = xHx^{−1}x = {^xH}x$ is also a left $^xH$-coset. Thus $k[xH]$ is in fact a $k^xH-kH$-bimodule, and for any…
scsnm
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Simple module $M$ over $R$ is isomorphic to $R/I$ where $I$ maximal

I know this's been asked here many times, and I now know a way to prove it using the first iso theorem on $\phi:R\to M$. But my first approach was the other way around and something was wrong: I tried to show that $f:M\to R/I$ is iso. More…
user642721
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Computation of $\hom_\mathbb{Z}(\mathbb{Z}_{p^\infty},\mathbb{Q})$

I'm trying to compute $\hom_\mathbb{Z}(\mathbb{Z}_{p^\infty},\mathbb{Q})$. I believe it is zero, simply because $\mathbb{Z}_{p^\infty}$ is torsion (it is a $p$-group) and $\mathbb{Q}$ is torsion-free. Is that argument correct? I'm suspicious of it…
user46225
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Example of a particular short exact sequence that does not split.

Can somebody give me an example of a short exact sequence of $\mathbb Z$-modules that starts with $\oplus_{i=1}^{\infty}\mathbb{Q}$ and does not split?
Robert
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How to show that if every cyclic left $R$-module is projective implies that for any left ideal $U$ of $R$, the left $R$-module $R/U$ is projective?

If every cyclic left $R$-module is projective, how can I show that for any left ideal $U$ of $R$, the left $R$-module $R/U$ is projective?
Bbbh
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Existence of non-zero subspace given linear map

Let $V$ be a real vector space over $\mathbb{R}$ with $\dim(V) \geq 3$. Show that $\exists W \subset V : W \ne V, W \ne \{0\}$ and $T(W) \subset W$ with $T \in End_{\mathbb{R}}(V)$ arbitrary. The solution considers $V$ as a $\mathbb{R}[T]$ module,…
user100101212
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