Proving that $\frac{R[x]}{I[x]} \cong \frac{R}{I} [x]$

Question

I define the map $$\phi : R[x] \to \frac{R}{I}[x], \quad \phi\left(\sum_{i=0}^{n} a_i x^i\right)=\sum_{i=0}^n \left(a_i+I\right)x^i $$ I know how to prove everything needed here except for showing that $\phi(a\cdot b)=\phi(a)\cdot\phi(b)$. I have seen other questions on this site regarding this but none of them addressed the part I'm stuck in. Please help. Thank you.

Answer 1

Hint:

By linearity, you only have to prove $\;\phi(ab)=\phi(a)\,\phi(b)$ for a pair of monomials.