For some polynomial $f(x)$ does the polynomial quotient ring $\mathbb{Z}_n [x] / f(x)$ always have finite cardinality?
Also, I have noticed that the examples of polynomial (over $\mathbb{Z}_n$) quotient rings that I have seen so far, if $f(x)$ has degree $d$ then $$ \mathbb{Z}_n [x] / f(x) = \left\{ [a_0 + a_1 x + a_2 x^2 + \dots + a_{d-1} x^{d-1}] \mid a_i \in \mathbb{Z}_n \right\} $$ Is this always true?