Graded Prüfer domains with Clifford homogeneous class semigroups

Abstract

Let $R=\underset{\alpha \in \Gamma }{\oplus }{R}_{\alpha }$ be an integral domain graded by an arbitrary torsionless commutative cancellative monoid Γ. We say that R is a graded Prüfer domain if each nonzero finitely generated homogeneous ideal of R is invertible. It is well known that if D is a Prüfer domain, then every nonzero locally principal ideal of D is invertible if and only if D is of finite character, if and only if D is a Clifford regular domain. In this paper, we generalize this result to graded Prüfer domains. That is, among other things, we prove that the following statements are equivalent for a graded Prüfer domain R; (i) each nonzero h-locally principal homogeneous ideal of R is invertible, (ii) each nonzero nonunit homogeneous element of R is contained in only finitely many maximal homogeneous ideals of R, and (iii) R is Clifford homogeneous regular. Let $D[\Gamma ]$ be the semigroup ring of Γ over an integral domain D and w be the so-called w-operation on $D[\Gamma ]$. We also study when $D[\Gamma ]$ is Clifford w-regular.

Keywords:

• Clifford homogeneous (w-)regular
• graded Prüfer domain
• PvMD

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A02
• 13A15
• 13F05

Acknowledgments

The authors would like to thank the referee for his/her careful reading and helpful suggestions which improved the original version of the paper.

Additional information

Funding

The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867).