On modules whose pure submodules are essential in direct summands

Abstract

We introduce the notion of \emph{pure extending modules}, a refinement of classical extending modules in which only pure submodules are required to be essential in direct summands. Fundamental properties and characterizations are established, showing that pure extending and extending modules coincide over von Neumann regular rings. As an application, we prove that pure extending modules admit decomposition patterns analogous to those in the classical theory, including a generalization of the Osofsky--Smith theorem: a cyclic module whose proper factor modules are pure extending decomposes into a finite direct sum of pure-uniform submodules. We establish sufficient conditions under which central quasi-morphicity and central morphicity coincide, notably for finitely generated, nonsingular, pure extending modules over noetherian rings, and for modules over semisimple artinian rings.