THE COHEN TYPE THEOREM FOR S-⁎ω-PRINCIPAL IDEAL DOMAINS

Let D be an integral domain, ${\ast}$ a star-operation on D, and S a (not necessarily saturated) multiplicative subset of D. In this article, we prove the Cohen type theorem for $S-{\ast}_{\omega}$-principal ideal domains, which states that D is an $S-{\ast}_{\omega}$-principal ideal domain if and only if every nonzero prime ideal of D (disjoint from S) is $S-{\ast}_{\omega}$-principal.