Ring Homomorphisms and Radical Ideals

Question

Let R$R$ and R′$R^{\prime }$ be commutative rings and let f:R→R′$f:R\to R^{\prime }$ be a ring homomorphism.
Let I$I$ and I′$I^{\prime }$ be ideals of R$R$ and R′$R^{\prime }$, respectively.

(a) Prove that f(I–√)⊂f(I)−−−−√$f(\sqrt{I})\subset \sqrt{f(I)}$.

(b) Prove that f−1(I′)−−−−−−√=f−1(I′−−√)$\sqrt{f^{-1}(I^{\prime })}=f^{-1}(\sqrt{I^{\prime }})$

(c) Suppose that f$f$ is surjective and ker(f)⊂I$\ker (f)\subset I$. Then prove that f(I–√)=f(I)−−









−−√