On the nil radical

Abstract

Let ${\sigma}$ be the class of all associative rings 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 n, m such that i ⁿ j ^m = 0, then I = 0 or J = 0. We will show that ${\sigma}$ is a special class and the upper radical ${\mathcal{U}(\sigma)}$ determined by ${\sigma}$ is equal to the nil radical ${\mathcal{N}}$ of Koethe. This implies that the nil radical ${\mathcal{N}(R)}$ of any ring R is the intersection of all ${\sigma}$-ideals I of R, that is, ideals I of R such that the factor ring R/I is in ${\sigma}$. We also give a necessary and sufficient condition for an ideal ${I \neq R}$ of a ring R to be a ${\sigma}$-ideal.