On Strongly $$J_n$$ J n -Clean Rings

Abstract

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 $$ea=ae$$ , $$a-e$$ is a unit and $$ea^n$$ 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 $$\pi $$ -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 $$2\times 2$$ matrix to be strongly $$J_2$$ -clean.