on a generalization of weak m-armendariz rings

For a ring r and a monoid m, we introduce j-m-armendariz rings which are a generalization of weak m-armendariz and j-armendariz rings, and we investigate their properties. we prove that if r / j(r) is a reduced ring and r is a j-m-armendariz ring, then r is j-m × 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 j-g-armendariz.