Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications - Jianwei He; Yajun Ma (Mathematics, 2025)
Let M be an Abelian monoid. A necessary and sufficient condition for the class <semantics>ArmM</semantics> of all Armendariz rings relative to M to coincide with the class <semantics>Arm</semantics> of all Armendariz rings is given. As a consequence, we prove that <semantics>ArmM</semantics> has exactly three cases: the empty set, <semantics>Arm</semantics>, and the class of all rings. If N is an Abelian monoid, then we prove that <semantics>ArmM×N=ArmM⋂ArmN</semantics>, which gives a partial affirmative answer to the open question of Liu in 2005 (whether R is <semantics>M×N</semantics>-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-<semantics>π</semantics>-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.