# Tagged Questions

1answer
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### $M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
1answer
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### Generators of equivalent rings

Let $A, B$ two rings. I know that $G \in \operatorname{mod}-A$ is a generator for $\operatorname{mod}-A$ if and only if $\operatorname{Hom}(G,-)$ is a faithful functor from $\operatorname{mod}-A$ to ...
2answers
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### Examples of Morita equivalent rings

Can someone give some examples of Morita equivalent rings different from the classical one? (i.e. that a ring $R$ is Morita equivalent to the ring $M_n(R)$)
3answers
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### Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers

Let $\hat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$. Since $\hat{\mathbb{Z}}$ is the inverse limit of the rings $\mathbb{Z}/n\mathbb{Z}$, it's a subgroup of \$\prod_n ...
0answers
78 views

### On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
1answer
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### Categorical definition of the characteristic of a ring

The characteristic of a ring is an important algebraic concept (and a specific number), but it refers to elements, so - in my understanding - it is evil (from the point of view of category theory). So ...
1answer
87 views

### Splitting idempotents

Let $C$ be an additive category. An idempotent $e=e^2\in\mathrm{Hom}_C(X,X)$ is split if there are morphisms $\mu:Y\rightarrow X$, $\rho:X\rightarrow Y$ such that $\mu\rho=id_Y$ and $\rho\mu=e$. ...
1answer
39 views

### Colimits of cosimplicial rings

The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings ...
3answers
86 views

### Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
1answer
183 views

### Center of a ring isomorphic to endomorphism ring of identity functor

I am having trouble verifying the following (this is self-study): There is an isomorphism between the center of a ring $A$ and the ring of endomorphisms of the identity functor of the category of ...
1answer
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### Inherited Morita similar rings

Let $R$ and $S$ be Morita similar rings. If a ring $R$ with the following property: every right ideal is injective. How do I prove that the ring $S$ has this property? If a ring $R$ with the ...
2answers
362 views

### What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
0answers
141 views

### Coproduct in the category of (noncommutative) associative algebras

For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
1answer
105 views

### Is the functor $R \mapsto \mathbb{M}_n(R)$ a right adjoint?

Given a positive integer $n$, there is a functor $F: \mathsf{Ring} \rightarrow \mathsf{Ring}$ such that $F(R) = \mathbb{M}_n(R)$ on objects and the action of $F$ on morphisms are given entrywise. Is ...
1answer
111 views

### On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
1answer
77 views

### Equivalent module categories

Let $A$ and $B$ be rings and let $A\text{-mod}$ and $B\text{-mod}$ be their abelian module categories. Let $F:A\text{-mod}\to M\text{-mod}$ and $F':B\text{-mod}\to A\text{-mod}$ be functors which ...
1answer
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### Does this “extension property” for polynomial rings satisfy a universal property?

On page 151 of Paolo Aluffi's Algebra: Chapter 0, an important property of the polynomial ring $\mathbb{Z}[x_1, \cdots, x_n]$ is introduced, namely that it's initial in the category of set functions ...
1answer
181 views

### Some isomorphism conditions

Here I want to ask some true or false problems regarding isomorphisms (and if false, is there some extra condition to make it true). I do not what is the correct title for these problem. And I also ...
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