Pairs of rings sharing their units
Gabriel Picavet, Martine Picavet L'Hermitte

TL;DR
This paper studies strongly local ring extensions where units are preserved, generalizing Cohn's rings, and explores their properties especially in J-regular rings, providing characterizations and examples.
Contribution
It introduces the concept of strongly local extensions, generalizes Cohn's rings, and characterizes these extensions in J-regular rings with new results and examples.
Findings
Characterization of strongly local extensions where units are preserved.
Complete description of J-regular strongly local extensions.
Equivalence of strongly local and weakly strongly inert properties when R is a field.
Abstract
We are working in the category of commutative unital rings and denote by the group of units of a nonzero ring . An extension of rings , satisfying is usually called local. This paper is devoted to the study of ring extensions such that , that we call strongly local. P. M. Cohn in a paper, entitled Rings with zero divisors, introduced some strongly local extensions. We generalized under the name Cohn's rings his definition and give a comprehensive study of these extensions. As a consequence, we give a constructive proof of his main result. Now Lequain and Doering studied strongly local extensions, where is semilocal, so that , where is the Jacobson radical of , is Von Neumann regular. These rings are usually called -regular. We establish many results on…
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Taxonomy
TopicsAdvanced Topics in Algebra · Rings, Modules, and Algebras
