In a recent paper, W. G.
Leavitt has called a radical class 𝒫 in a universal class 𝒲 of not necessarily
associative rings strongly hereditary if 𝒫(I) = I ∩𝒫(R) for all ideals I of any ring
R ∈𝒲. In this paper, strongly hereditary radicals are investigated and a new
construction is provided for the minimal strongly hereditary radical containing a
given class in 𝒲. Nonassociative versions of some results of E. P. Armendariz on
semisimple classes are proved, including a characterization of semisimple classes
corresponding to strongly hereditary radicals.
