Question #17354

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

Expert's answer

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.

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