On monoid rings constructed by reduced rings

Let R be a ring and F [ f0g the free monoid generated by U = fu1; : : : ; utg with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, Mn = 0. In this note, we show that the monoid ring R[M] is Armendariz if and only if R is reduced.