One-sided duo property on nilpotents

Abstract

We study the structure of nilpotents 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) Köthe'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.