Graded Prüfer domains with Clifford homogeneous class semigroups

Abstract

Let 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 be the semigroup ring of Γ over an integral domain D and w be the so-called w-operation on . We also study when is Clifford w-regular.