A note on β-classes

A ring [Formula: see text] is an essential extension of its ideal [Formula: see text] if [Formula: see text], for every nonzero ideal [Formula: see text] of [Formula: see text]. Let [Formula: see text] be the prime radical. A class [Formula: see text] of prime rings that is closed under ideals and essential extensions is called a [Formula: see text]-class if [Formula: see text], where [Formula: see text] is the upper radical determined by [Formula: see text]. Let * be the class of all semiprime rings [Formula: see text] such that [Formula: see text] [Formula: see text] for every ideal [Formula: see text] of [Formula: see text] with [Formula: see text]. In this paper, we give some necessary and sufficient conditions for [Formula: see text] to be a [Formula: see text]-class. Intersections of [Formula: see text]-classes have always been found to be [Formula: see text]-classes and Leavitt asked whether this is always true. We show some of the strong restrictions which must apply to any possible counterexample.