Is $\frac{R[x]}{(f(x))}$ finite for $R$ finite and $f$ not monic polynomial?

Question

If $R$ is a finite commutative ring, then is $\frac{R[x]}{(f(x))}$ finite with $f$ not monic polynomial?

I can prove above claim if f(x) is monic polynomial using division algorithm? But I am not possible further as if f(x) is not monic polynomials then I can not use division algorithm.

Any Help will be appreciated

Answer 1

In $\Bbb Z_4[x]/(2x)$, the elements $1, x, x^2, x^3, \ldots$ are all distinct.