On quasi-S-multiplication modules and weak S-multiplication modules

Let [Formula: see text] be a multiplicatively closed subset of a commutative ring with identity [Formula: see text]. In this paper, we introduce quasi-[Formula: see text]-multiplication modules and weak [Formula: see text]-multiplication modules which are generalizations of quasi-multiplication modules and weak multiplication modules. Our aim is to establish some basic results on the above-mentioned notions and to generalize some well-known results on [Formula: see text]-multiplication modules, quasi-multiplication modules and weak multiplication modules to quasi-[Formula: see text]-multiplication modules and weak [Formula: see text]-multiplication modules.