On Monoid Rings Over Nil Armendariz Ring

Given a ring R and a monoid M, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ring R[M] instead of the polynomial ring R[x]. We prove that several properties transfer between R and the monoid ring R[M], in case R is nil M-Armendariz ring. We resolve the structure of nil M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil M-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nil M-Armendariz, for any unique product monoid M. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nil M-Armendariz rings behaves under various ring extensions.