Question

Show that rad R is the smallest ideal I ⊆ R such that R/I is J-semisimple.

Answer 1

Since if an ideal I ⊆ R is such that R/I is J-semisimple, then I⊇ rad R. (The J-semisimplicity of R/I means that the intersection of the maximal left ideals of R containing I is exactly I. It follows that rad R, the intersection of all the maximal left ideals of R, is contained in I.) Then R is the smallest ideal I ⊆ R such that R/I is J-semisimple.