Lifting of automorphisms of factor modules

ABSTRACT

This paper introduces the notion of dual of automorphism extendable modules. A module M is called automorphism-extendable if for every submodule N of M, every automorphism of N can be extended to an endomorphism of M. We call a module M a dual automorphism-extendable module if whenever K is a submodule of M, then every automorphism ν:M∕K → M∕K lifts to an endomorphism 𝜃 of M. In this paper we give various examples of dual automorphism-extendable modules and study their properties. In particular, we prove that every dual automorphism-extendable module is a D3-module. It is shown that over a right artinian ring R, an R-module $M={\oplus }_I{M}_i$ with hollow modules M_i is dual automorphism-extendable if and only if M is quasi-projective.

KEYWORDS:

• Artinian ring
• dual automorphism-extendable module
• dual automorphism-invariant module
• perfect ring
• quasi-projective module

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D40
• 16D80
• 16N80

Acknowledgments

The authors would like to thank the referees for the very helpful comments and suggestions.