Open AccessArithmetic Rings and their generalizations
Author(s) -
Rosario Strano
Publication year - 2018
Publication title -
bollettino dell'accademia gioenia di scienze naturali
Language(s) - English
Resource type - Journals
ISSN - 0393-7143
DOI - 10.35352/gioenia.v51i381.1
Subject(s) - mathematics , generalization , unit (ring theory) , commutative ring , ring (chemistry) , arithmetic , commutative property , domain (mathematical analysis) , pure mathematics , integral domain , discrete mathematics , algebra over a field , field (mathematics) , mathematical analysis , mathematics education , chemistry , organic chemistry
Prüfer domains are characterized by various properties regarding ideals and operations between them. In this note we consider six of these properties. The natural generalization of the notion of Prüfer domain to the case of a commutative ring with unit, not necessarily a domain, is the notion of arithmetic ring. We ask if the previous properties characterize arithmetic ring in the case of a general commutative ring with unit. We prove that four of such properties characterize arithmetic rings while the remaining two are weaker and give rise to two different generalizations.