# Questions tagged [modules]

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

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### Is the tensor product of a free module with any other module free? [closed]

Let $A$ be a ring and $F, M$ two $A$-modules such that $F$ is free. Is then the tensor product $M \otimes_A F$ free? I know that the tensor product is free if both of them are free $A$-modules. But ...
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### Why $eAe$ is identical with $\mathrm{End}(eA)$ [duplicate]

Let $A$ be an algebra and $e$ is an (maybe full) idempotent. I saw there is an isomophism between $eAe$ and $\mathrm{End}(eA)$. How is it constructed？
1 vote
25 views

### Tower of module-endomorphism rings

Let $R$ be a ring with nonzero left ideal $A$. Define $E_1=\text{End}({}_RA)$ viewed as a ring of right operators on $A$ and $E_2=\text{End}(A_{E_1})$ viewed as a ring of left operators on $A$. ...
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1 vote
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### Double Centralizer Property in simple ring without identity

In Lam's book, $\textit{A first course in noncommutative rings}$, Theorem 3.11 states: Let $R$ be a simple ring, and $A$ be a nonzero left ideal. Let $D = End({}_RA)$ (viewed as a ring of right ...
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• 743
1 vote
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### When are Idempotents elements of a semisimple algebra primitive

Let $A=KG$ be a $K$-algebra such that $|G| \in K^{\times}$. Here $A$ is a semisimple algebra. Consider the decomposition of $A$ into simple components:$$A=A_1 \times A_2 \times \cdots \times A_k.$$ ...
• 2,331
104 views

### Number of non-equivalent irreducible representations of a finite group $G$ over an arbitrary field $F$/non-isomorphic simple $F[G]$-modules

in my algebra class, they give us as an exercise to prove that a finite group $G$ admits at most finitely many non-equivalent irreducible representations over an arbitrary field $F$. Now, I showed ...
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