# Questions tagged [modules]

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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### Subcategory determined by composition series

Suppose $A$ is an artin algebra and take the category $\operatorname{mod}A$ of finitely generated $A$-modules. Consider the following construction. Let $M$ be an $A$-module. Since $A$ is artinian, $M$ ...
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### Is being finitely generated a local property

Searching on this site and others leads to lots of dicussion about localisation at multiplicatively closed subsets of the form $\{f_i^j\}_{j=1}^\infty$ where $\{f_i\}_{i=1}^n$ generate the whole ring ...
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### When is $\mathbb Z/n\mathbb Z$ semisimple as $\mathbb Z$-module? [duplicate]

I'm looking for some characterization of $\mathbb Z/n\mathbb Z$ being semisimple as $\mathbb Z$-module. Obviously for $n$ prime this is the case, as $\mathbb Z/n\mathbb Z$ is simple under that ...
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How can i understand of the free module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$ where $e_i=(0,\ldots ,1, \ldots 0) \in K[x]^r$ denotes the i–th canonical basis vector of $K[x]^r$. We call $x^\alpha e_i=(0,... 1 vote 0 answers 41 views ### Every finitely generated module is sum of local modules Note: All modules are over a$K$-algebra$A$with$K$a field and the underlying ring of$A$is unital (but not necessarily commutative). Definition: A module$V$is local if there is a maximal ... -4 votes 1 answer 54 views ### Free module over a countable set [closed] Suppose$R$is a commutative ring. Is$R^{\oplus \mathbb{N}}$isomorphic to$R^{\oplus \mathbb{N} }\oplus R^{\oplus \mathbb{N}}$as$R$-modules? If so, how do I find an explicit isomorphism? Edit: I ... 1 vote 1 answer 47 views ### Why direct sum of modules admits canonical projections? This is more of a moral (i.e. category theoretical) question. In the category of$R$-modules for a ring$R$, the product is the direct product$M=\prod_{i\in I}M_i$with canonical projections$\pi_i\...
Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection. I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting ...