π-Armendariz rings relative to a monoid

Let M be a monoid. A ring R is called M-π-Armendariz if whenever α = a 1 g 1 + a 2 g 2 + · · · + a n g n , β = b 1 h 1 + b 2 h 2 + · · · + b m h m ∈ R[M] satisfy αβ ∈ nil(R[M]), then a i b j ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.