Quasi-Armendariz rings relative to a monoid

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