The monoid of regular elements in commutative rings with zero divisors

Let R be a commutative ring with identity, R center dot be the multiplicative monoid of regular elements in R, t be the so-called t-operation on R or R center dot. 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 center dot. 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 center dot)) be the set of regular prime t-ideals of R (resp., the set of non-empty prime t-ideals of R center dot), and Cl(A) be the class group of A for A = R or R center dot. Then, among other things, we prove that the map phi:t-spec(R)-> t-spec(R center dot) given by phi(P)=P center dot is bijective; Cl(R) approximately equal to Cl(R center dot); and R is a factorial ring if and only if R center dot is a factorial monoid.