4
votes
0answers
58 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
1
vote
0answers
42 views

Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
1
vote
0answers
43 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
1
vote
1answer
118 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
5
votes
1answer
69 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
9
votes
2answers
152 views

What kind of object is the kernel of a ring homomorphism?

The category $\mathbf{Grp}$ of groups has a zero object, namely the trivial group $1$. Since $\mathbf{Grp}$ is furthermore complete, we have the notion of a kernel of a group homomorphism. The kernel ...
6
votes
1answer
106 views

Are there categorifications of prime or irreducible elements (of a ring, say)?

I'm very sorry if this is a duplicate in any way or is otherwise a stupid question. I've looked around (for quite a while) but . . . no luck. There's a categorification of what it means to be an ...
2
votes
1answer
72 views

Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ ...
13
votes
0answers
119 views

Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
2
votes
1answer
60 views

Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
0
votes
0answers
38 views

Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
5
votes
0answers
63 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
3
votes
1answer
58 views

In a Morita context ($A, B, P, Q, f, g$), why are $P$ and $Q$ projective if $f$ is surjective?

The title probably says it all :). This is probably a very very simple question, please bear with me. Let $(A, B, P, Q, f, g)$ be a Morita context (or pre-equivalence data) as defined by Hyman Bass in ...
3
votes
0answers
43 views

Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
0
votes
1answer
48 views

Coproducts and direct products

Is the arbitrary direct sum of modules a submodule of their coproduct? Ie is $\underset{i \in I}{\coprod} M_i \cong \underset{i \in I}{\bigoplus} M_i$... if not then if each $M_i$ where to be ...
11
votes
1answer
136 views

$B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
6
votes
1answer
85 views

The infinite Direct Sum in the category Ring

If you don't have strong personal feelings about it already, most of you have at least witnessed the opposing factions on how we should define a ring and, by extension, how we should define a ring ...
4
votes
1answer
78 views

How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
0
votes
2answers
39 views

Proof that the localization $R_S$ is naturally isomorphic to the localization at the saturation $R_{\overline{S}}$?

Localizations have the universal property that if $S$ is a multiplicative subset of a commutative ring $R$, and $i\colon S\to R$ is the canonical embedding, then if $g\colon R\to T$ is any map such ...
7
votes
1answer
134 views

How to construct the coproduct of two (non-commutative) rings

How to construct/describe the coproduct of two - not necessarily commutative - rings $R$ and $S$? This in category $\mathbf{Rng}$ having as objects rings with a unit and as arrows unitary ...
6
votes
1answer
74 views

How to look at a polynomial ring based on a ring that is not commutative?

When I first met polynomial rings $R[X]$ I wondered: 'where do they come from?' Later the idea that - if $R$ is commutative - they could be interpreted as $R$-algebras free over a singleton brought ...
2
votes
1answer
59 views

Is there a special name for ring homomorphisms $f : R \rightarrow S$ with $f^*(C(R)) \subseteq C(S)$?

Edit. For some reason, I called the functor $F$ described below a full functor as opposed to a faithful functor. The problem has now been corrected. For any ring $R$, let $C(R)$ denote the center of ...
6
votes
0answers
87 views

Lie algebra of Derivations as a functor?

To an associative algebra $A$ one can associate a Lie algebra $\operatorname{Der} A$ of all derivations $D:A\to A$. To any morphism of associative algebras $\alpha:A\to B$, how can one associate a ...
3
votes
0answers
60 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
-1
votes
2answers
117 views

Polynomial ring and the free algebra

In the Algebra book of Mac Lane there is an exercise in Chap. IV which tells me to construct a polynomial ring $A[X]$ for any set (not necessarily finite) $X$ ($A$ a ring), and to give correct the ...
4
votes
3answers
264 views

How to construct polynomial ring $K[x]$ over commutative ring $K$ by making use of universal arrows.

In CWM of Mac Lane I encounter: the construction of a polynomial ring $K\left[x\right]$ in an indeterminate $x$ over a commutative ring $K$ is a universal construction. Unfortunately this ...
0
votes
1answer
55 views

Colimit of Reduced rings

Let R be the comlimit $\operatorname {lim} R_i$ of rings $R_i$. Let $R_{ired}$ denote the quotient ring $R_i/I_{nil}$ where $I_{nil}$ is the ideal of nilpotent elements. What is $\operatorname {lim} ...
3
votes
2answers
95 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
1
vote
0answers
63 views

Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
4
votes
0answers
156 views

What does it mean for a ring to be unital?

What is the category of unital rings like? I only know that it no more has a terminal object. But what about the products and coproducts? Are they as usual, different or nonexistent? In Gelfand ...
5
votes
1answer
145 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
0
votes
1answer
65 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
27 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 ...
4
votes
3answers
291 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 $M_n(R)$)
14
votes
3answers
938 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 ...
4
votes
0answers
240 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
146 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
186 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
49 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
183 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
342 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
49 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 ...
20
votes
2answers
587 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 ...
8
votes
2answers
386 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
123 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 ...
8
votes
1answer
360 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
89 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
317 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
271 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 ...