Answer to Question #17354 in Abstract Algebra for sanches

Question #17354
Show that rad R is the smallest ideal I ⊆ R such that R/I is J-semisimple.
1
Expert's answer
2012-10-31T08:53:48-0400
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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