2025/03/10 by Gunay, Gulizar, Mermut, Engin
#Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2503.07271
A module M is called stable if it has no nonzero projective direct summand. For a ring R , we study conditions under which R-modules from certain classes decompose as a direct sum of a projective submodule and a stable submodule. Over an arbitrary ring, modules of finite uniform dimension or finite hollow dimension can be decomposed as a direct sum of a projective submodule and a stable submodule. By using the Auslander-Bridger transpose of finitely presented modules, we prove that every finitely presented right R-module over a left semihereditary ring R has such a decomposition. Our main focus in this article is to give examples where such a decomposition fails. We give some ring examples over which there exists an infinitely generated or finitely generated or finitely presented module where such a decomposition fails. Our main example is a cyclically presented module M over a commutative ring such that~M has no such decomposition and M is not projectively equivalent to a stable module.