When is a 2 × 2 Matrix Ring Over a Commutative Local Ring Quasipolar?

Abstract

A ring R is quasipolar if for any a ∈ R, there exists p ² = p ∈ R such that p ∈ comm²(a), p + a ∈ U(R) and ap ∈ R ^qnil . In this article, we determine when a 2 × 2 matrix over a commutative local ring is quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be quasipolar. Consequently, we obtain several equivalent conditions for the 2 × 2 matrix ring over a commutative local ring to be quasipolar. Furthermore, it is shown that the 2 × 2 matrix ring over the ring of p-adic integers is quasipolar.

Key Words:

• Commutative local ring
• Matrix ring
• Quasipolar ring
• Strongly clean ring

2000 Mathematics Subject Classification:

• 16U99
• 15A09
• 16S99

ACKNOWLEDGMENTS

The authors would like to express their gratitude to the referee for correcting many errors and valuable suggestions. This research is supported by the National Natural Science Foundation of China (10871042, 10971024), and the Specialized Research Fund for the Doctoral Program of Higher Education (200802860024).

Notes

Communicated by S. Sehgal.