For a monoid M, we introduce M-nil-Armendariz rings, which are generalizations of nil-Armendariz rings and M-Armendariz rings, and we investigate their properties. We show that every NI ring is M-nil-Armendariz for any unique product monoid M, and if R is a 2-primal and M-Armendariz ring, then R is M?× N-nil-Armendariz, where N is a unique product monoid. Moreover, we study the relationship between the weak annihilator ideals of a ring R and those of the monoid ring R[M] in case R is M-nil-Armendariz.
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