P. F. Smith [7, Theorem 8] gave sufficient conditions on a finite set of modules for their sum and intersection to be multiplication
modules. We give sufficient conditions on an arbitrary set of multiplication modules for the intersection to be a multiplication
module. We generalize Smith"s theorem, and we prove conditions on sums and intersections of sets of modules sufficient for
them to be multiplication modules.
Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).