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.

Keywords

• right (left) nilpotent-duo ring
• nilpotent
• right (left) duo ring
• radical
• NI ring
• matrix ring
• ascending chain condition on left (right) annihilators

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