Abstract
Let R be a commutative ring with identity, be the multiplicative monoid of regular elements in R, t be the so-called t-operation on R or
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
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
Let R be a t-Marot ring, t-
(resp., t-
) be the set of regular prime t-ideals of R (resp., the set of non-empty prime t-ideals of
), and
be the class group of A for A = R or
Then, among other things, we prove that the map
-
-
given by
is bijective;
and R is a factorial ring if and only if
is a factorial monoid.
2020 Mathematics Subject Classification:
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.