Investigation of a class of associative rings P which is a radical class in the sense of Kurosch. (Kuroch, A.G. Radicals in rings and algebras. Mat. Sb. N.S. Moscow, 1953, 33(75): 13-26.). P is the class of rings every maximal ideal of which contains some power of the ring. The possibility of P having the hereditary property is discussed along with its position in the chain of prominent radical classes.