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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.

## A meditation on semiadditive categories

The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, $\text{Set}$-enriched).

## Finite noncommutative probability, the Born rule, and wave function collapse

The previous post on noncommutative probability was too long to leave much room for examples of random algebras. In this post we will describe all finite-dimensional random algebras with faithful states and all states on them. This will lead, in particular, to a derivation of the Born rule from statistical mechanics. We will then give a mathematical description of wave function collapse as taking a conditional expectation.

## Internal equivalence relations

For the last few weeks I’ve been working as a counselor at the PROMYS program. The program runs, among other things, a course in abstract algebra, which was a good opportunity for me to get annoyed at the way people normally introduce normal subgroups, which is via the following unmotivated

Definition: A subgroup $N$ of a group $G$ is normal if $gNg^{-1} \subset N$ for all $g \in G$.

It is then proven that normal subgroups are precisely the kernels $N = \phi^{-1}(e)$ of surjective group homomorphisms $\phi : G \to G/N$. In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.

Today I’d like to present an alternate conceptual route to this definition starting from equivalence relations and quotients. This route also leads to ideals in rings and, among other things, highlights the special role of the existence of inverses in the theory of groups and rings (in the latter I mean additive inverses). The categorical setting for this discussion is the notion of a kernel pair and of an internal equivalence relation in a category, but for the sake of accessibility we will not use this language explicitly.

## ab, ba, and the spectrum

Let $a, b$ be two $n \times n$ matrices. If $a, b$ don’t commute, then $ab \neq ba$; however, the two share several properties. If either $a$ or $b$ is invertible, then $ab$ is conjugate to $ba$, so in particular they have the same characteristic polynomial.

What if neither $a$ nor $b$ are invertible? As it turns out, $ab$ and $ba$ still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have $ab = 0$ but $ba$ nonzero). There are several ways of proving this result, which implies in particular that $ab$ and $ba$ have the same eigenvalues.

What if $a, b$ are linear transformations on an infinite-dimensional vector space? Do $ab$ and $ba$ still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even $\mathbb{Z}$ fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one with the property that every element $r \in R$ acts nontrivially in some simple (left) module $M$.

Said another way, let the Jacobson radical $J(R)$ of a ring consist of all elements of $r$ which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring $R$ is then semiprimitive if it has trivial Jacobson radical.

The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s A first course in noncommutative rings.

## Morita equivalence and the bicategory of bimodules

In the previous post we learned that it is possible to recover the center $Z(R)$ of a ring $R$ from its category $R\text{-Mod}$ of left modules (as an $\text{Ab}$-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings $R, S$ are said to be Morita equivalent if the categories $R\text{-Mod}, S\text{-Mod}$ are equivalent as $\text{Ab}$-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.

## Centers, 2-categories, and the Eckmann-Hilton argument

The center $Z(G)$ of a group is an interesting construction: it associates to every group $G$ an abelian group $Z(G)$ in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor $\text{Grp} \to \text{Ab}$ (unlike the abelianization $G/[G, G]$). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category $C$ we may associate to an object $c \in C$ its automorphism group $\text{Aut}(c)$ or endomorphism monoid $\text{End}(c)$), and this is a canonical construction which again doesn’t extend in an obvious way to a functor $C \to \text{Grp}$ or $C \to \text{Mon}$. (It merely reflects some special part of the bifunctor $\text{Hom}(-, -)$.)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group $G$, but of the identity functor $G \to G$, where $G$ is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.