Tagged Questions

0
votes
1answer
50 views

$M\cong_{R\operatorname{-Mod}}N$ 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 ...
1
vote
1answer
21 views

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 ...
2
votes
2answers
84 views

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 $M_n(R)$)
7
votes
3answers
255 views

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 ...
2
votes
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 ...
4
votes
1answer
93 views

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 ...
3
votes
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 $\rho\mu=e$. ...
2
votes
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 ...
1
vote
3answers
86 views

Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
3
votes
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 ...
5
votes
1answer
41 views

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 ...
18
votes
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 ...
3
votes
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 ...
2
votes
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 ...
3
votes
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? ...
1
vote
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 ...
4
votes
1answer
222 views

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 ...
4
votes
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 ...