Answer to Question #17277 in Algebra for sanches

Question

Show that if f : R → S is a surjective ring homomorphism, then f(rad R) ⊆ rad S.
Give an example to show that f(rad R) may be smaller than rad S.

Answer 1

The inclusion f(rad R)⊆ rad S is clear since, for any maximal left ideal m of S,the inverse image f^−1(m) is a maximal left idealof R. To give an example of strict inclusion, let R = Z and let fbe the natural projection of R onto S = Z/4Z. Here, f(radR) = f(0) = 0, but rad S = 2Z/4Z is nonzero.