Let $R_1$ and $R_2$ be rings, $\phi:R_1\to R_2$ some ring homomorphism. Consider the map $\widehat\phi$ that sends any polynomial $f(x) = \sum_ia_ix^i\in R_1[x]$ to $\widehat\phi(f(x)) = \sum_i\phi(a_i)x^i\in R_2[x]$.
Is $\widehat\phi$ a ring homomorphism? Clearly it respects addition, but I am having trouble verifying that it respects multiplication.
For $f(x):=\sum_i a_ix^i$ and $g(x):=\sum_j b_jx^j$, we have $f(x)g(x)=\sum_k\sum_{i+j=k}a_ib_jx^k$, thus
$$\begin{align}\widehat\phi[f(x)g(x)]&=\sum_k\sum_{i+j=k}\phi(a_ib_j)x^k\\&=\sum_k\sum_{i+j=k}\phi(a_i)\phi(b_j)x^k\\&=[\sum_i \phi(a_i)x^i][\sum_j\phi(b_j)x^j]=\ldots\end{align}$$