On Commutative FGS-Rings

Abstract

Let R be a ring. An R-module M is said to satisfy property (S) if every surjective endomorphism of M is an automorphism. Let F _R be the class of finitely generated R-module and S _R be the class of R-module satisfying property (S). In 1969 W. V. Vasconcelos proved that for every commutative ring R, F _R  ⊆ S _R . In general this inclusion is strict. In this note we characterize commutative ring R for which F _R  = S _R .

Key Words:

• FGS-ring
• Principal ideal ring
• Finitely generated module

Acknowledgments

The authors would like to thank Professor W. V. Vasconcelos and the referee for their valuable suggestions. With their comments many proofs have been simplified and they have helped us to improve considerably the first version of the paper.

Notes

^#Communicated by W. Vasconcelos.