Rings whose cyclic modules are lifting and \(\oplus\)-supplemented. (English) Zbl 1400.16002
Summary: 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 \(\oplus\)-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.
MSC:
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 16D80 | Other classes of modules and ideals in associative algebras |
| 16L30 | Noncommutative local and semilocal rings, perfect rings |
| 16P20 | Artinian rings and modules (associative rings and algebras) |
Keywords:
Artinian serial ring; cyclic module; direct sum of local modules; lifting module; projective cover preserving direct summands; \(\oplus\)-supplemented moduleReferences:
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