One-sided duo property on nilpotents

Abstract

We study the structure of nilpotent elements in relation with a ring property that is near to one-sided duo rings. Such a property is said to be one-sided nilpotent-duo. We prove the following for a one-sided nilpotent-duo ring R: (i) The set of nilpotents in R forms a subring; (ii) Kothe's conjecture holds for R; (iii) the subring generated by the identity and the set of nilpotents in R is a one-sided duo ring; (iv) if the polynomial ring R[x] over R is one-sided nilpotent-duo then the set of nilpotents in R forms a commutative ring, and R[x] is an NI ring. Several connections between one-sided nilpotent-duo and one-sided duo are given. The structure of one-sided nilpotent-duo rings is also studied in various situations in ring theory. Especially we investigate several kinds of conditions under which one-sided nilpotent-duo rings are NI.