Reflexive Property of Rings

Abstract

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

Key Words:

• Matrix ring
• Polynomial ring
• Reflexive ring
• Right (left) idempotent reflexive ring
• Right principally quasi-Baer ring
• Right quotient ring
• Semiprime ring

2000 Mathematics Subject Classification:

• 16S99
• 16U80

ACKNOWLEDGMENTS

The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much. They also thank Professor Jin Yong Kim for raising problems for idempotent reflexive rings.

This work was supported by the Daejin University Special Research Grants in 2008.

Notes

Communicated by J. L. Gomez Pardo.