Small supplements weak supplements and proper classes

Abstract

Let SS denote the class of short exact sequences E:0 → Af→ B → C → 0 of R-modules and R-module homomorphisms such that f(A) has a small supplement in B i.e. there exists a submodule K of M such that f(A) + K = B and f(A) ∩ K is a small module. It is shown that SS is a proper class over left hereditary rings. Moreover in this case the proper class SS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplement submodules. The homological objects such as SS-projective and SScoinjective modules are investigated. In order to describe the class SS we investigate small supplemented modules i.e. the modules each of whose submodule has a small supplement. Besides proving some closure properties of small supplemented modules we also give a complete characterization of these modules over Dedekind domains. © 2020 Elsevier B.V. All rights reserved.