On the nil radical (Q343219)

In this article, a special class of \(\sigma\) associative rings is given. Explicitly, \(\sigma = \{R\mid\) if \(I\) and \(J\) are ideals of \(R\), and for all \(i\in I\) and \(j\in J\) there are some natural numbers \(m\) and \(n\) such that \(i^nj^m = 0\), then either \(I = (0)\) or \(J = (0)\}\). The upper radical \(\mathcal U(\sigma)\) determined by \(\sigma\) is equal to the nil radical \(\mathcal N\) of Koethe, and consequently, one sees that the nil radical \(\mathcal N(R)\) of any ring \(R\) is the intersection of all \(\sigma\)-ideals of \(R\). Here, a \(\sigma\)-ideal \(I\) of a ring \(R\) is one that the factor ring \(R/I\in\sigma\). It is also shown that a proper ideal \(I\) of a ring \(R\) is a \(\sigma\)-ideal if and only if the complement \(R\backslash I\) of \(I\) in \(R\) contains a complete system \(S\) such that \((r) \cap (s) \cap S\neq \emptyset\) for any \(r, s \in R\backslash I\).