On Modules Whose Pure Submodules Are Essential in Direct Summands
Kaushal Gupta, Theophilus Gera, Amit Sharma, Ashok Ji Gupta
TL;DR
The paper introduces pure extending modules as a homological refinement of extending modules, tying the concept to purity and direct-sum decompositions via $C_1$-inspired structure. It also defines RD-pure extending modules to capture element-wise purity, establishing hierarchy, Morita invariance, and closure properties. A key contribution is a decomposition result: a cyclic module whose cyclic factor modules are pure extending decomposes into a finite direct sum of pure-uniform submodules, generalizing the Osofsky–Smith theorem under purity. Additionally, the work connects these notions to morphic module theory, proving both a negative general answer to whether every centrally quasi-morphic module is centrally morphic and positive results under finite generation, nonsingularity, and purity hypotheses, thereby enriching the interaction between purity, endomorphism structures, and decomposition theory.
Abstract
We introduce the notion of 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. Additionally, we resolve an open problem of Dehghani and Sedaghatjoo by constructing a centrally quasi-morphic module that is not centrally morphic, arising from the link between pure-extending behavior and nonsingularity in finitely generated modules over Noetherian rings.