Quasi-normality of idempotents on nilpotents

Abstract

We study the structure of idempotents in non-Abelian rings, concerning a ring property near to the normality of idempotents on the set of nilpotents. We call a ring with such property right idempotent-quasi-normalizing on nilpotents (simply, right IQNN), and study the structure of right IQNN rings in relation with matrix rings, polynomial ring, and factor rings, by which we extend the class of right IQNN rings. It is proved that the class of IQNN rings contains the 2 by 2 full matrix rings over fields and the upper triangular matrix rings over reduced rings. It is shown that given any countable field K, there exists a semiprime IQNN algebra R over K such that the polynomial ring R[x] over R is IQNN but not NI, and the upper nilradical of R[x] is zero.