A New Class of Small Submodules

Let R be a commutative ring with identity 1≠ 0 and M a nonzero unital R-module. In this paper, we introduce a new notion of submodules in M, namely T-semi-annihilator small submodules of M with respect to an arbitrary submodule T of M. A submodule N of M is T-semi-annihilator small in M provide that for each submodule X of M with T⊆ N+X implies that Ann(X) (T:M). In addition, we investigate some results concerning to this new class of submodules. Among various results, we prove that for a faithful finitely generated multiplication module M, the submodule N of M is a T-semi-annihilator small submodule of M if and only if $(N:M)$ is a $(T:M)$-semi-annihilator small ideal of R. Finally, we explore the properties and the behaviour of this structure under ring homomorphism, localization, direct sums and tensor product of them with a faithfully flat R-module.