Homomorphsim preimage of a principal ideal still principal

Question

If $\phi : R \rightarrow R'$ is a homomorphism, is it true that $(\phi^{-1}(a))=\phi^{-1}((a))$? Thanks

Answer 1

No, e.g. for $(x)\subset \mathbb{Z/2 Z}[x]$ and $\varphi:\mathbb{Z}[x]\to\mathbb{Z/2 Z}[x]$ you get $\varphi^{-1}((x))=(2,x)$.