Rings whose nilpotents form a multiplicative set

ABSTRACT

Nilpotents in Armendariz and abelian π-regular rings are multiplicatively closed. However, this fact need not hold in many kinds of rings. This article concerns a class of rings whose nilpotents are closed under multiplication. Rings contained in this class are said to be nilpotent-closed, and the structure of nilpotent-closed rings is investigated in relation with various situations which happen ordinarily in the study of noncommutative ring theory. In the procedure, various sorts of rings are investigated, so that they are nilpotent-closed. We also study familiar conditions under which nilpotents in nilpotent-closed rings form a subring.

KEYWORDS:

• Abelian ring
• form a subring
• matrix ring
• multiplicative set
• NI ring
• nil-IFP ring
• nilpotent
• nilpotent-closed ring
• polynomial ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U80
• 16N40

Acknowledgments

The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much.