Ideal chains in radical theory. (Q949831)

One of the most useful characterizations of a radical class \(\mathcal R\) of rings is a non-empty homomorphically closed class which is closed under extensions and satisfies the condition (C) If \(\{I_\lambda\mid\lambda\in\Lambda\}\) is a chain of ideals of some ring and \(I_\lambda\in\mathcal R\) for all \(\lambda\in\Lambda\), then \(\bigcup_{\lambda\in\Lambda}I_\lambda\in\mathcal R\). In this paper the author shows that closure under unions of chains of ideals can be replaced by closure under unions of continuous well-ordered chains of ideals or by some other variants of this condition. Some consequences also are discussed.