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.
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Expert's answer
2012-10-30T10:27:29-0400
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.
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