On domain-like objects in the category of unitary rings

W.D. Burgess ; R. Raphael

Theory and applications of categories, Tome 41 (2024), pp. 927-959

Résumé

In the category C of unitary rings, Barr, Kennison and Raphael (2015) studied the limit closures of various classes of commutative integral domains; in particular the class of all domains and of integrally closed domains to form the reflective subcategories K_dom and K_ic. This article looks at K_dom(R) and K_ic(R) - the same symbols are used for the subcategories and the reflection functors - for some rings R. The objects in K_dom can be called "domain-like". Of particular interest is the ring C^1(R), the ring of real functions with continuous first derivative. The ring L is that of continuous functions that are C^1 on a dense open set of R. Then K_dom(C^1(R)) ⊂ K_ic(C^1(R)) ⊂ L, with proper inclusions. Moreover, for f in K_dom(C^1(R)), f has one-sided derivatives and those are one-sided continuous.

For commutative rings R ⊂ S in R there is a subring of the integral closure of R in S introduced here, called the split integral closure, that is explored and turns out to be very useful.

Détail

Publié le : 2024-08-09

Classification : 13G05, 13B22, 18A35, 26A15
Keywords: commutative semiprime rings, domain objects, limit closures, $\Co$ functions