Give an example of a finite ring $R$ and polynomials $f, g\in R[X]$ such that the polynomial division of $f$ by $g$ is not unique

Question

Give an example of a finite ring $R$ and polynomials $f, g\in R[X]$ such that the polynomial division of $f$ by $g$ is not unique.

I know that $f$ can be written as $\sum_na_nX^n$ and $g$ can be written as $\sum_mb_mX^m$, and I know that $\deg(f+g)\leq\max(m,n)$, and $\deg(f\cdot g)\leq \deg(f)+\deg(g)$.

But I don't know how to work on this problem. Please any help.

Answer 1

To get this off the unanswered queue,

$2x^2=2(x^2+2)$ in $\mathbb{Z}_4[x]$.

Answer by Bill Dubuque

To add on to this, ring with zero divisors provide many examples. For instance, when working with $\mathbb{Z}_n[x]$, there are always such polynomials when $n$ is not a prime.