On S-strong Mori domains

Let D be an integral domain, S be a (not necessarily saturated) multiplicative subset of D, w be the so-called w-operation on D, and M be a unitary D-module. As generalizations of strong Mori domains (respectively, UFDs) and strong Mori modules, we define D to be an S-strong Mori domain (respectively, S-factorial domain) if for each nonzero ideal I of D  , there exist an s∈Ss∈S and a w-finite type (respectively, principal) ideal J of D   such that sI⊆J⊆IwsI⊆J⊆Iw; and M to be an S-strong Mori module if M is a w-module and for each nonzero submodule N of M  , there exist an s∈Ss∈S and a w-finite type submodule F of N   such that sN⊆F⊆NwsN⊆F⊆Nw. This paper presents some properties of S-strong Mori domains, S-factorial domains and S-strong Mori modules.