Quasi-Armendariz rings relative to a monoid

For a monoid M, we introduce M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings, and investigate their properties. The M-quasi-Armendariz condition is a Morita invariant property. The class of M-quasi-Armendariz rings is closed under some kinds of upper triangular matrix rings. Every semiprime ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M. Moreover, we study the relationship between the quasi-Baer property of a ring R and those of the monoid ring R[M]. Every quasi-Baer ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M.