# Answer to Question #17277 in Algebra for sanches

Question #17277

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.

Give an example to show that f(rad R) may be smaller than rad S.

Expert's answer

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*f*be the natural projection of*R*onto*S*= Z*/*4Z. Here,*f*(rad*R*) =*f*(0) = 0*,*but rad*S*= 2Z*/*4Z is nonzero.Need a fast expert's response?

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