On a Generalization of the π-Armendariz Condition

We in this note consider a new concept, so called M-π-Armendariz rings, which is a generalization of both π-Armendariz rings and M-Armendariz rings. We show that every NI ring is M-π-Armendariz for any unique product monoid M. We also claim that if R is both a 2-primal ring and an M-Armendariz ring, then R is (M × N)-π-Armendariz, where N is a unique product monoid