On $\star$-Semi-homogeneous Integral Domains

Abstract

Let $\star$ be a finite character star-operation defined on an integral domain D. A nonzero finitely generated ideal of D is $\star$-homogeneous if it is contained in a unique maximal $\star$-ideal. And D is called a $\star$-semi-homogeneous ($\star$-SH) domain if every proper nonzero principal ideal of D is a $\star$-product of $\star$-homogeneous ideals. Then D is a $\star$-semi-homogeneous domain if and only if the intersection D $=$ $\underset{P\in \star \text {-}{\text {Max}}(D)}{\bigcap D_{P}}$ is independent and locally finite where $\star$-${\text {Max}}(D)$ is the set of maximal $\star$-ideals of D. The $\star$-SH domains include h-local domains, weakly Krull domains, Krull domains, generalized Krull domains, and independent rings of Krull type. We show that by modifying the definition of a $\star$-homogeneous ideal we get a theory of each of these special cases of $\star$-SH domains.