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Non-unital rings

(This post was originally intended to go up immediately after the sequence on Gelfand duality.)

A rng (“ring without the i”) or non-unital ring is a semigroup object in $\text{Ab}$. Equivalently, it is an abelian group $A$ together with an associative bilinear map $m : A \otimes A \to A$ (which is not required to have an identity). This is what some authors mean when they say “ring,” but this does not appear to be standard. A morphism between rngs is an abelian group homomorphism which preserves multiplication (and need not preserve a multiplicative identity even if it exists); this defines the category $\text{Rng}$ of rngs (to be distinguished from the category $\text{Ring}$ of rings).

Until recently, I was not comfortable with non-unital rings. If we think of rings either algebraically as endomorphisms of abelian groups or geometrically as rings of functions on spaces, then there does not seem to be any reason to exclude the identity endomorphism resp. the identity function on a space. As for morphisms which don’t preserve identities, if $X \to Y$ is any map between spaces of some kind, then the identity function $Y \to F$ ($F$ is, say, a field) is sent to the identity function $X \to F$, so not preserving identities when they exist seems unnatural.

However, not requiring or preserving identities turns out to be natural in the theory of C*-algebras; in the commutative case, it corresponds roughly to thinking about locally compact Hausdorff spaces rather than just compact Hausdorff spaces. In this post we will discuss rngs generally, including a discussion of the geometric picture of commutative rngs, to get more comfortable with them. It turns out that we can study rngs by formally adjoining multiplicative identities to them. This is an algebraic version of taking the one-point compactification, and it allows us to extend Gelfand duality, in a suitable sense, to locally compact Hausdorff spaces (see this math.SE question for the precise statement, which we will not discuss here).

Regular and effective monomorphisms and epimorphisms

Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.

In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.

Z[sqrt{-3}] is the Eisenstein integers glued together at two points

Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by $\text{Spec } R$ I mean a commutative ring $R$ regarded as an object in the opposite category $\text{CRing}^{op}$.