Article
| Title: | Decompositions into a direct sum of projective and stable submodules (English) |
| Author: | Günay, Gülizar |
| Author: | Mermut, Engin |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 787-807 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | projective submodule |
| Keyword: | stable module |
| Keyword: | Auslander-Bridger transpose |
| Keyword: | finitely presented module |
| Keyword: | decomposition of module |
| MSC: | 13F05 |
| MSC: | 16D50 |
| MSC: | 16D60 |
| DOI: | 10.21136/CMJ.2026.0079-25 |
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| Date available: | 2026-09-11T13:11:28Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153728 |
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