Let $ R $ be a domain and $ f \in R[X] $ be a monic polynomial. Suppose there exists an isomorphism $ \phi $ between $ R[X]/(f) $ and $ R \times R $. Then show that:
i) If $ \phi(X)=(a,b) \in R \times R $ then $ f(a)=f(b)=0 $.
ii) If $ p(X) \in R[X] $ such that $ \phi(p(X))=(1,0) $ then $ p(a)=1 $ and $ p(b)=0 $.
I tried to show at i) that $ (X-a)(X-b) | f(X) $ so I wrote $ f(X)=(X-a)(X-b)q(X)+Ax+B $ and we must show that $ A=B=0 $. We know that $ \phi (f(X))=(0,0) $ so we get that $$ \phi(X-a)\phi(X-b)\phi(q(X))+\phi(A)\phi(X)+\phi(B)=(0,0) $$ Hence $$ ((a,b)-\phi(a))((a,b)-\phi(b))\phi(q(X))+\phi(A)(a,b)+\phi(B)=(0,0) $$ and I don't know how to go on from this and I also don't know how to approach ii).
I would appreciate any help. Thank you!