Quasipolar Property of Generalized Matrix Rings

Abstract

The article concerns the question of when a generalized matrix ring K _s (R) over a local ring R is quasipolar. For a commutative local ring R, it is proved that K _s (R) is quasipolar if and only if it is strongly clean. For a general local ring R, some partial answers to the question are obtained. There exist noncommutative local rings R such that K _s (R) is strongly clean, but not quasipolar. Necessary and sufficient conditions for a single matrix of K _s (R) (where R is a commutative local ring) to be quasipolar is obtained. The known results on this subject in [5] are improved or extended.

Key Words:

• Generalized matrix ring
• Local ring
• Matrix ring
• Quasipolar ring
• Strongly clean ring
• Strongly π-regular ring

2010 Mathematics Subject Classification:

• Primary: 16U99, 16S99
• Secondary: 16L30, 16S50

ACKNOWLEDGMENTS

The authors are grateful to the referee for valuable comments and suggestions. Part of the work was carried out when the first author was visiting the Memorial University of Newfoundland.

Notes

Communicated by S. Sehgal.