Decompositions into a direct sum of projective and stable submodules

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.