In the Amitsur-Kurosch theory of radicals in rings (2), an important problem is to determine the relationship between the radical of a ring and the radical of each of its ideals. The first result on this problem was by Amitsur who proved that if β is a hereditary radical in the sense that ideals of β-radical rings are β-radical, then for each associative ring R and ideal I of R, β(I) = I ∩ β(R), where β(R) denotes the β-radical of R; see (2).

Later, Suliński, Divinsky, and the author proved that if β is any radical and R is an associative or alternative ring, then β(I) ⊆ I∩ β(R) for each ideal I of R; see (3). Since every hereditary radical β has the property β(I) ⊇ I ∩ β(R), this result provided another proof of Amitsur's theorem and extended that theorem to alternative rings. Of course, this raises the question of whether Amitsur's theorem is true for Lie or Jordan rings.