On Strongly Jn-Clean Rings.

A ring R is said to be strongly J n -clean if n is the least positive integer such that every element a is strongly J n -clean, that is, there exists an idempotent e such that e a = a e , a - e is a unit and e a n is in the Jacobson radical J(R). It is proved that strongly J n -clean ring is a strongly clean ring with stable range one. If R is an abelian ring (a ring in which all idempotents are central), then R is strongly J n -clean if and only if R / J(R) is strongly π -regular and idempotents lift modulo J(R). Some examples and basic properties of these rings are studied. Some criterions in terms of solvability of the characteristic equation are obtained for such a 2 × 2 matrix to be strongly J 2 -clean. [ABSTRACT FROM AUTHOR]