Jacobson categories (Q1335109)

As its title indicates, the main purpose of this paper is to introduce and study the notion of Jacobson category. These categories find their origin in the properties of the category of algebraic commutative algebras over a field \(k\). Typically, this category has the property that inverting an element in an algebra is equivalent to annihilating another element, which may be chosen to be idempotent. This, and many other features, are to be found within any Jacobson category. The main axiom of Jacobson categories asserts the existence of enough finitely presentable directly codisjunctable objects, i.e., which give rise to ``objects of fractions'', which are direct factor morphisms. It is proven that Jacobson categories are Zariski categories, introduced and studied by the same author. In fact, they are precisely the Zariski categories in which any object is a Jacobson object, i.e., any prime congruence on any object is a meet of maximal congruences. Other characterizations and examples are provided as well. The last part of the paper concentrates on Stone semi-algebras and, in particular, it is proved that the set of finitely generated congruences on any object of a Jacobson category is a Stone semi-algebra, this providing still another characterization of Jacobson categories.