The monoid of regular elements in commutative rings with zero divisors

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 $\ge 2.$ Let R be a t-Marot ring, t-$\text{spec}(R)$ (resp., t-$\text{spec}(R^{•})$) 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 $\phi :t$-$\text{spec}(R)\to t$-$\text{spec}(R^{•})$ given by $\phi (P)=P^{•}$ is bijective; $Cl(R)\cong Cl(R^{•});$ and R is a factorial ring if and only if $R^{•}$ is a factorial monoid.

Keywords:

• Class group
• Krull monoid
• Krull ring
• rank-one DVR
• t-Marot ring
• t-operation

2020 Mathematics Subject Classification:

• 13A15; 13C20; 13F05; 20M12

Acknowledgments

The authors would like to thank the anonymous referee for his/her very careful reading, helpful comments, and useful suggestions to include a weakly Krull ring (in particular, Lemma 4.6 that have eventually led to the formulation of Proposition 4.7 and Example 4.8), which improved the original version of this paper greatly.

Additional information

Funding

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867).