On the saturations of submodules

Abstract

Let R ⊆ S be a ring extension, and let A be an R-submodule of S. The saturation of A (in S) by τ is set A[τ] = {x ∈ S : tx ∈ A for some t ∈ τ}, where τ is a multiplicative subset of R. We study properties of saturations of R-submodules of S. We use this notion of saturation to characterize star operations ⋆ on ring extensions R ⊆ S satisfying the relation (A ∩ B)⋆ = A⋆ ∩ B⋆ whenever A and B are two R-submodules of S such that AS = BS = S.