Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications

Let M be an Abelian monoid. A necessary and sufficient condition for the class ArmM of all Armendariz rings relative to M to coincide with the class Arm of all Armendariz rings is given. As a consequence, we prove that ArmM has exactly three cases: the empty set, Arm, and the class of all rings. If N is an Abelian monoid, then we prove that ArmM×N=ArmM⋂ArmN, which gives a partial affirmative answer to the open question of Liu in 2005 (whether R is M×N-Armendariz if R is M-Armendariz and N-Armendariz). We also show that the other Armendariz-like rings relative to an Abelian monoid, such as M-quasi-Armendariz rings, skew M-Armendariz rings, weak M-Armendariz rings, M-π-Armendariz rings, nil M-Armendariz rings, upper nil M-Armendariz rings and lower nil M-Armendariz rings can be handled similarly. Some conclusions on these classes have, therefore, been generalized using these classifications.