Pairs of rings sharing their units
Gabriel Picavet\,^1 and Martine Picavet-L’Hermitte\,^2
\,^{1,2} Math̩matiques 8 Rue du Forez, 63670 РLe Cendre France.
Pages 207-238 | Received 12 November 2024, Accepted 21 March 2025, Published 26 January 2026
Abstract
We are working in the category of commutative unital rings and denote by \mathrm U(R) the group of units of a nonzero ring R. An extension of rings R\subseteq S, satisfying \mathrm U(R)=R \cap\mathrm U(S) is usually called local. This paper is devoted to the study of ring extensions such that \mathrm U(R)=\mathrm U(S), 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 S is semilocal, so that S/\mathrm J(S), where \mathrm J(S) is the Jacobson radical of S, is Von Neumann regular. These rings are usually called J-regular. We establish many results on J-regular rings in order to get substantial results on strongly local extensions when S is J-regular. The Picard group of a J-regular ring is trivial, allowing to evaluate the group \mathrm U(S)/\mathrm U(R) when R is J-regular. We then are able to give a complete characterization of the Doering-Lequain context. A Section is devoted to examples. In particular, when R is a field, the strongly local and weakly strongly inert properties are equivalent.
Keywords: Group of units, local extension, strongly local extension, J-regular ring, integral extension, FCP extension
MSC numbers: Primary 13B02; Secondary 13B25.
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