2024/11/04 by Gabriel Pıcavet, Picavet, Gabriel, Martine Picavet L'Hermitte +1
Mathematics · #13B02 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2411.02101
openalex publication_date 2024/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊆ S, satisfying \mathrm U(R)=R ∩\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.