## Ring Homomorphisms and Radical Ideals

## Problem 624

Let $R$ and $R’$ be commutative rings and let $f:R\to R’$ be a ring homomorphism.

Let $I$ and $I’$ be ideals of $R$ and $R’$, respectively.

**(a)** Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$.

**(b)** Prove that $\sqrt{f^{-1}(I’)}=f^{-1}(\sqrt{I’})$

**(c)** Suppose that $f$ is surjective and $\ker(f)\subset I$. Then prove that $f(\sqrt{I}\,) =\sqrt{f(I)}$