On Strongly J-Clean Rings

Abstract

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T _n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation.

Key Words:

• Local ring
• 2 × 2 Matrix
• Strongly J-clean ring

2000 Mathematics Subject Classification:

• 16S50
• 16U99

ACKNOWLEDGMENTS

The author is grateful to the referee for his/her suggestions which helped me to improve the presentation considerably.

Notes

Communicated by V. A. Artamonov.