Rings whose cyclic modules are lifting and ⊕-supplemented

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.

KEYWORDS:

• Artinian serial ring (with square-zero radical)
• cyclic module
• direct sum of local modules
• lifting module
• projective cover preserving direct summands
• ⊕-supplemented module

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D40
• 16D80
• 16L30

Acknowledgments

This research constitutes part of the Ph.D dissertation of Nguyen at Memorial University of Newfoundland. He acknowledges the support by a Vietnamese Government Fellowship and a Fellowship from Memorial University.