Another characterization of the upper nil radical

Abstract

In 2015 Halina France-Jackson introduced the notion of a ${\sigma}$-ring i.e. a ring R with the property that if I and J are ideals of R and for all ${i\in I}$, ${{j\in J}}$, there exist natural numbers m, n such that ${i^{m}j^{n} =0}$, then I = 0 or J = 0. It is shown that ${\sigma}$ is a special class which coincides with the class ${\rho}$ of all prime nil-semisimple rings. This implies that the upper nil radical of any ring R is the intersection of all ideals I of the ring such that R/I is a ${\sigma}$-ring. In this paper we introduce classes of rings equivalent to the ${\sigma}$-rings and then give characterizations of the upper nil radical in terms of these rings.