# 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

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.

• $2xx = 2x(x+2)$ in $\,\Bbb Z_4[x].\$ That should give you an idea of how to generale many more exmples. – Gone Nov 26 '19 at 16:57
• @BillDubuque Why are you answering in a comment? – Arthur Nov 26 '19 at 17:04
• @Arthur So the OP can learn and give an answer and receive feedback on it. – Gone Nov 26 '19 at 17:07
• @BillDubuque Sure. But you can do that just as well, or even better, in an answer post. – Arthur Nov 26 '19 at 17:09
• Because $2(2) = 0$ in $\Bbb Z_4.\$ Both $x$ and $x+2$ are valid quotients for $\,2x^2 \div 2x\,$ so the quotient is not unique. – Gone Nov 26 '19 at 18:03

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