A Class of Quasipolar Rings

Abstract

An element a of a ring R is called J-quasipolar if there exists p ² = p ∈ R satisfying p ∈ comm²(a) and a + p ∈ J(R); R is called J-quasipolar in case each of its elements is J-quasipolar. The class of this sort of rings lies properly between the class of uniquely clean rings and the class of quasipolar rings. In particular, every J-quasipolar element in a ring is quasipolar. It is shown, in this paper, that a ring R is J-quasipolar iff R/J(R) is boolean and R is quasipolar. For a local ring R, we prove that every n × n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) ≅ ℤ₂. Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 × 2 matrix over a commutative local ring is J-quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be J-quasipolar.

Key Words:

• J-quasipolar ring
• Local ring
• Quasipolar ring
• Triangular matrix ring
• Uniquely clean ring

2010 Mathematics Subject Classification:

• 16U99
• 15A09
• 16S99

ACKNOWLEDGMENTS

The authors are highly grateful to the referee for correcting many errors and valuable comments. This research was supported by the National Natural Science Foundation of China (10971024, 11201064), and the Natural Science Foundation of Jiangsu Province (BK2010393).

Notes

Communicated by S. Sehgal.