# Tagged Questions

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### Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
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### Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
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### Minimal Frattini Extensions

From "Construction of Finite Groups" article : "To construct minimal Frattini extensions of $H$ we have to consider all suitable irreducible $H$-modules $M$. Clearly, $M$ can be viewed as an ...
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### Extension of a group by a module

What is the definition of an extension of a group by a module? and what is the difference between extensions by groups and modules?
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### What does this notation mean, which looks like direct sum?

I ran across this notation in a paper. What does $Z^{\bigoplus n(n-1)}$ mean? I knew that $Z^n$ or $Z^{(n)}$ means direct sum, but never met this one.
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### What does this notation mean? Perhaps related to abelian group and module.

I found this in a paper, but I don't it's meaning. Where H is an abelian group, what does $\bigwedge ^{2} H$ mean? If H is a dual vector space, perhaps it is wedge product. But I don't know when it ...
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### Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
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### How to show a Hom group is isomorphic to some modulo group?

Let $\mathbb{Z}_{n}[m]=\{a\in{\mathbb{Z}_{n}|ma=0\}}$, where $m$ is a positive integer. Then how do we establish the following? $$\mathbb{Z}_{n}[m]\cong{\mathbb{Z}_{\gcd(m,n)}}$$
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### Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
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### Isomorphism theorem for Abelian groups, related to Hatcher exercise 2.1.14

I am trying to understand which Abelian groups can fit the short exact sequence $$0 \rightarrow \mathbb{Z}_{p ^m}\rightarrow A \rightarrow \mathbb{Z}_{p^n}\rightarrow 0.$$ ...
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### How do we compute $\mathbb{Z}^2/(n, m)$?

I have been trying to compute quotients of this form (as modules). Does anyone have a quick method of doing this? I'm sure I'm missing something obvious. Any help is appreciated!
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### Why are projective modules cohomologically trivial?

Let $G$ be a finite group, $H\subset G$ a subgroup, $k$ a commutative ring, $M$ a $kG$-module, $n\in\mathbb{Z}$, and $\hat{H}\,^n(H,M)$ the $n$th Tate cohomology group as defined in this question, ...
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### Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
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### Hungerford Algebra - Chapter IV - proof of the splitting lemma

Im working with the splitting lemma right now and I don't understand it too much. If someone has that book and could help me complete the sketch in page 177, proof of 1.18) I would be very grateful. ...