Weakly automorphism invariant modules and essential tightness

ABSTRACT

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.

KEYWORDS:

• Automorphism-invariant module
• CEP-ring
• Goldie ring
• QF-ring
• tight module
• weakly-injective module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D50
• 16U60
• 16W20

Acknowledgments

We thank the referee for his/her careful reading of the paper. The second author was supported by the National Foundation for Science and Technology Development of Vietnam, Gebze Technical University and the Russian and Government Program of Competitive Growth of Kazan Federal University.