Homomorphsim preimage of a principal ideal still principal

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

• $\phi^{-1}(a)$ is not a single element in general. – lhf Oct 2 '18 at 12:22
• adding to @lhf and if it is not the claim is false. $\phi:R^2\rightarrow R$ with $\phi(x,y) = x$, choose any non-trivial $R$ you want. – Yanko Oct 2 '18 at 12:23

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