On Nil-semicommutative Rings

Weixing Chen

Abstract

In this note a ring R is defined to be nil-semicommutative in case for any a, b ∈ R, ab is nilpotent implies that arb is nilpotent whenever r ∈ R. Examples of such rings include semicommutative rings, 2-primal rings, NI-rings etc.. It is proved that if I is an ideal of a ring R such that both I and R/I are nil-semicommutative then R is nil-semicommutative and that if R is a semicommutative ring satisfying the \alpha-condition for an endomorphism of R then the skew polynomial ring R[x; \alpha] is nil-semicommutative. However the polynomial ring R[x] over a nil-semicommutative ring R need not be nil-semicommutative. It is an open question whether a nil-semicommutative ring is an NI-ring, which has a close connection with the famous Koethe’s conjecture.