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