Decompositions into a direct sum of projective and stable submodules
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| Format: | Preprint |
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2025
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| _version_ | 1866910095265234944 |
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| author | Gunay, Gulizar Mermut, Engin |
| author_facet | Gunay, Gulizar Mermut, Engin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decompositions into a direct sum of projective and stable submodules Gunay, Gulizar Mermut, Engin Commutative Algebra Rings and Algebras 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. |
| title | Decompositions into a direct sum of projective and stable submodules |
| topic | Commutative Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2503.07271 |