On radicals and polynomial rings

Abstract

For any class $\mathcal{M}$ of rings, it is shown that the class ${\mathcal{E}}_{\ell }(\mathcal{M})$ of all rings each non-zero homomorphic image of which contains either a non-zero left ideal in $\mathcal{M}$ 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 $\gamma$ under certain constraints, one can obtain a strictly decreasing chain of radicals $\gamma ={\gamma }_{(1)}\supset {\gamma }_{(2)}\supset \cdot \cdot \cdot \supset {\gamma }_{(n)}\supset \cdot \cdot \cdot$ where, for each positive integer $n$, $\gamma (n)$ is the radical consisting of all rings $A$ such that $A[x_1,\dots ,x_n]$ is in $\gamma$, thus giving a negative answer to a question posed by Gardner. Moreover, classes $\mathcal{M}$ of rings are constructed such that there exist several such radicals $\gamma$ in the interval $[{\mathcal{E}}_{\ell }(0),{\mathcal{E}}_{\ell }(\mathcal{M})]$ .