Regular ideals in commutative rings, sublattices of regular ideals, and Prüfer rings

Abstract

AbstractIn this paper we investigate regularly generated, regular, semiregular, and faithful ideals in a commutative ring R and the sublattices they determine. Connections with multiplicative lattice theory are given. Given a Prüfer ring R we show that there is a Prüfer domain D with the sublattice of regular ideals of R isomorphic to the lattice of ideals of D. Numerous examples of rings with zero divisors having certain properties are given. A Prüfer ring with an invertible ideal that is not generated by regular elements is constructed. An example is given to show that the intersection of two regular principal ideals need not be generated by regular elements