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Rings whose cyclic modules are lifting and ⊕-supplemented.
- Source :
- Communications in Algebra; 2018, Vol. 46 Issue 11, p4918-4927, 10p
- Publication Year :
- 2018
-
Abstract
- It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules. [ABSTRACT FROM AUTHOR]
- Subjects :
- RING theory
MODULES (Algebra)
MATHEMATICAL proofs
GROUP theory
ABSTRACT algebra
Subjects
Details
- Language :
- English
- ISSN :
- 00927872
- Volume :
- 46
- Issue :
- 11
- Database :
- Complementary Index
- Journal :
- Communications in Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 131906826
- Full Text :
- https://doi.org/10.1080/00927872.2018.1459646