Quasi-normality of idempotents on nilpotents

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