Nilradicals of the unique product monoid rings

Abstract

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.