On δ-semiperfect modules

ABSTRACT

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules.

KEYWORDS:

• δ-Lifting module
• δ-semiperfect module
• projective δ-cover
• semiperfect module

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D99
• 16E50
• 16L30

Notes

¹This terminology has been used differently in [12].

²This notion has been termed as δ-supplemented module in [3], [7] and [14].

³This terminology has been used differently in [3].