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Keywords:
projective submodule; stable module; Auslander-Bridger transpose; finitely presented module; decomposition of module
projective submodule; stable module; Auslander-Bridger transpose; finitely presented module; decomposition of module
Summary:
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 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.
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