A CLASS OF QUASIPOLAR RINGS

作者：Cui, Jian; Chen, Jianlong^*

来源：Communications in Algebra, 2012, 40(12): 4471-4482.

DOI：10.1080/00927872.2011.610854

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

An element a of a ring R is called J-quasipolar if there exists p(2) = p is an element of R satisfying p is an element of comm(2)(a) and a + p is an element of 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 x n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) congruent to Z(2). Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 x 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.

• 出版日期2012
• 单位安徽师范大学; 东南大学