# Tagged Questions

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### The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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### Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
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### Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
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### Intuitive Explanation of Hom Functor Property

I am new to category theory and keep running across it while studying algebra. Whenever I see the $\mathrm{Hom}$ functor mentioned (in the context of modules), two of its basic properties are listed: ...
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### Does “maximal submodule <=> simple quotient module” generalize to abelian categories?

Does the statement "If $A$, $B$ are modules over a commutative ring $R$, then $B$ is a maximal submodule of $A$ if and only if $A/B$ is a simple module" generalize to the setting of abelian ...
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### A example of a monoidal non symmetric category of $R$-bimodules

It is well know that if $R$ is a commutative ring, then the category of modules $_R\operatorname{Mod}$ is monoidal symmetric. (Edit. I wrote the category of $R$-bimodules, as F. Muro explained this ...
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### What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
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### Questions about Rickards proof that $D^b_\mathtt{sg}(A) \equiv \mathtt{stmod}(A)$

Setup: Let $A$ be a self-injective algebra (so projective = injective for modules) and let $D^b(A)$ and $K^b(A)$ be the bounded derived category and the full subcategory consisting of the perfect ...
### Can we really understand $R$ by studying $R$-modules? [duplicate]
According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...