On $\ast $-Semi Homogeneous Domains

Let $\ast$ be a finite character star operation defined on an integral domain $D.$ Call a nonzero $\ast$-ideal $I$ of finite type a $\ast$ -homogeneous ($\ast$-homog) ideal, if $I\subsetneq D$ and $(J+K)^{\ast }\neq D$ for every pair $D\supsetneq J,K\supseteq I$ of proper $\ast$ -ideals of finite type$.$ Call an integral domain $D$ a $\ast$-Semi Homogeneous Domain ($\ast$-SHD) if every proper principal ideal $xD$ of $D$ is expressible as a $\ast$-product of finitely many $\ast$-homog ideals. We show that a $\ast$-SHD contains a family $\mathcal{F}$ of prime ideals such that (a) $D=\cap_{P\in \mathcal{F}}D_{P},$ a locally finite intersection and (b) no two members of $\mathcal{F}$ contain a common non zero prime ideal. The $\ast$-SHDs include h-local domains, independent rings of Krull type, Krull domains, UFDs etc. We show also that we can modify the definition of the $\ast$-homog ideals to get a theory of each special case of a $\ast$-SH domain.