On Nil – Semi Clean Rings

Abstract

In this paper, the notions of semi-idempotent elements and nil-semiclean elements are introduced. We show that, if a ring R is a nil-semi clean ring, then the Jacobson radical, J(R), of R is nil. We also characterize some properties related to orthogonal idempotent elements e and f in a ring R. We prove that if R is a nil-semi clean ring and the idempotent elements of R are Central, then the Jacobson radical J(R) of R equals the set of all nilpotent elements of R. Finally, we show that if R is a commutative ring, then the polynomial ring R[x] is not nil-semi clean ring.

Keywords: Idempotent elements; clean elements; Semi clean elements.

2000 Mathematics Subject Classification. Primary: 16N40, Secondary: 16N20