On Jacobson and nil radicals related to polynomial rings. (Q2813360)

Recall, a ring \(R\) is called Armendariz if \(f_ig_j=0\) for any coefficients \(f_i\) and \(g_j\) of \(f(x)\in R[x]\) and \(g(x)\in R[x]\), respectively, whenever \(f(x)g(x)=0\). These, and related rings, have already been studied extensively. Here the authors continue such investigations and look at the polynomial ring over such rings as well as the properties of the polynomial ring when the factor ring by its Jacobson radical is Armendariz. This latter consideration led them to introduce \textit{feckly Armendariz} rings. This is a ring for which the factor ring by its Jacobson radical is Armendariz. They are, in particular, interested in the Jacobson and nil radicals of polynomial rings.NEWLINENEWLINE The connections between these and related types of rings are determined and many examples are given to show when relationships are sharp. For example, an Armendariz ring is in general not a feckly Armendariz ring and a polynomial ring over an Armendariz ring is feckly Armendariz. If \(R\) is feckly Armendariz, then \(R[x]\) need not be feckly Armendariz, but the power series ring \(R[[x]]\) will always be.