Armendariz rings relative to a monoid

摘要

For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings, and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid At. We show that if R is a reduced and M-Armendariz ring, then R is M x N-Armendariz, where N is a unique product monoid. It is also shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.