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On domain-like objects in the category of unitary rings

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W.D. Burgess, R. Raphael

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.

Keywords:
commutative semiprime rings, domain objects, limit closures, $\Co$ functions

2020 MSC:
13G05, 13B22, 18A35, 26A15

*Theory and Applications of Categories,*
Vol. 41, 2024,
No. 29, pp 927-959.

Published 2024-08-09.

http://www.tac.mta.ca/tac/volumes/41/29/41-29.pdf

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