Residual submodules of multiplication modules

Let R be a commutative ring with identity and M an R-module. We introduce and give some properties and characterizations of the concepts of Mcancellation, M-weak cancellation, M-meet principal, and M-weak meet principal ideals. We prove that a submodule of a faithful multiplication module is faithful (resp. finitely generated, multiplication, flat, projective, pure, prime) if and only if its residual by a finitely generated faithful multiplication ideal is a faithful (resp. finitely generated, multiplication, flat, projective, pure, prime) submodule.