The monoid of regular elements in commutative rings with zero divisors
Gyu Whan Chang, Jun Seok Oh
Abstract
Let R be a commutative ring with identity, R• be the multiplicative monoid of regular elements in R, t be the so-called t-operation on R or R•. A Marot ring is a ring whose regular ideals are generated by their regular elements. Marot rings were introduced by J. Marot in 1969 and have been playing a key role in the study of rings with zero divisors. The notion of Marot rings can be extended to t-Marot rings such that Marot rings are t-Marot rings. In this paper, we study some ideal-theoretic relationships between a t-Marot ring R and the monoid R•. We first construct an example of a t-Marot ring that is not Marot. This also serves as an example of a rank-one DVR of reg-dimension ≥2. Let R be a t-Marot ring, t-spec(R) (resp., t-spec(R•)) be the set of regular prime t-ideals of R (resp., the set of non-empty prime t-ideals of R•), and Cl(A) be the class group of A for A = R or R•. Then, among other things, we prove that the map φ:t-spec(R)→t-spec(R•) given by φ(P)=P• is bijective; Cl(R)≅Cl(R•); and R is a factorial ring if and only if R• is a factorial monoid.