The Armendariz property on ideals

In the present note we study the Armendariz property on ideals of rings, introducing a new concept which unifies the Armendariz property and the insertion-of-factors-property (simply, IFP) for rings. In relation with this work, we investigate rings over which polynomial rings are IFP, called strongly IFP rings, which generalize both ideal-Armendariz rings and strongly reversible rings. The classes of minimal noncommutative ideal-Armendariz rings and strongly IFP rings, and the classes of minimal non-Abelian ideal-Armendariz rings and strongly IFP rings are completely determined, up to isomorphism. It is also shown that a local ring is Armendariz, symmetric, and strongly reversible (hence ideal-Armendariz) when the cardinality of the Jacobson radical is 4.