Decompositions into a direct sum of projective and stable submodules
Czechoslovak Mathematical Journal, vol.76, no.3, pp.787-807, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 76 Issue: 3
- Publication Date: 2026
- Doi Number: 10.21136/cmj.2026.0079-25
- Journal Name: Czechoslovak Mathematical Journal
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET, Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Page Numbers: pp.787-807
- Keywords: Auslander-Bridger transpose, decomposition of module, finitely presented module, projective submodule, stable module
- Dokuz Eylül University Affiliated: Yes
Abstract
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.