On radicals and polynomial rings

Abstract

For any class ℳ of rings, it is shown that the class ℰℓ(ℳ) of all rings each non-zero homomorphic image of which contains either a non-zero left ideal in ℳ or a proper essential left ideal is a radical. Some characterizations and properties of these radicals are presented. It is also shown that, for radicals γ under certain constraints, one can obtain a strictly decreasing chain of radicals γ = γ(1) ⊃ γ(2) ⊃ ··· ⊃ γ(n) ⊃ ··· where, for each positive integer n, γ(n) is the radical consisting of all rings A such that A[ x1, … ,xn ] is in γ, thus giving a negative answer to a question posed by Gardner. Moreover, classes ℳ of rings are constructed such that there exist several such radicals γ in the interval [ ℰℓ(0),ℰℓ(ℳ) ] .