# Questions tagged [modules]

For questions on modules over rings.

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### If $R$ is a regular square-free ring, then $R$ is dual-square-free

Let $R$ be a ring with unity. We call $R$ regular if every principal (left) right ideal is a summand (i.e., generated by an idempotent). A module $M$ is called square-free if whenever $A,B$ are ...
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### A question on clean rings

Recall that a ring $R$ is called clean if every element of $R$ is a sum of a unit of $R$ and an idempotent of $R$. We call a module $M$ clean if its endomorphism ring $End(M)$ is a clean ring. ...
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### How to find the annihilator of a module/how to determine the torsion module

Given a left $R$-module $M$. Given a subset of $S\subset M$. What are related theorems concerning the annihilator of $S\subset M$ that can be used in practice? $S$ is known. In my case, $R$ is ...
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### When is a submodule trivial?

I am a beginner concerning module theory, but I need it for my PhD. Sorry for obvious questions therefore. Given a left $C(G)$-module $(V, \tilde{\rho})$ where $C(G)$ denotes the group algebra over a ...
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### Do graded-commutative rings satisfy the strong rank condition?

Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$. It is ...
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### Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
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### The change-of-monoid adjunction between categories of modules induced by a morphism of monoids

Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...
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### Schur functors = Weyl functors in characteristic zero?

I asked this question on Math Stack Exchange https://math.stackexchange.com/questions/4789924/schur-functors-weyl-functors-in-characteristic-zero, but I got no answers, so I ask the same question here....
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### When is the intersection of a family of CS-modules is again CS?

Recall that a right module $M$ over a ring $R$ (with unity) is called CS if every submodule of $M$ is essential in a summand of $M$. Let $\lbrace M_i \rbrace_{i\in I}$ be a family of right CS-modules ...
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### When is a module a filtered colimit of finitely presented submodules?

For a (commutative, say) ring $R$, and an $R$-module $M$ it is known that $M$ is both: a filtered colimit of finitely generated $R$-submodules (by considering all finite subsets of $M$ and ...
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### A general theory of pairings

Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra. There are also text books for bilinear forms and related quadratic ...
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### Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)

Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
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### Trivial group cohomology induces trivial cohomology of subgroups

From the answer to another question I asked (Projective representations of a finite abelian group) and from the structure theorem of finite abelian groups it follows that if $A$ is a finite abelian ...
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### Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?

I'm confused about some nomenclature in a paper by J. Goguen from 1975. I'm happy to use definitions however they appear. But I aim to summarise the paper for a non-expert audience and I want them to ...
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