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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.

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    $\begingroup$ $2xx = 2x(x+2)$ in $\,\Bbb Z_4[x].\ $ That should give you an idea of how to generale many more exmples. $\endgroup$ – Gone Nov 26 '19 at 16:57
  • $\begingroup$ @BillDubuque Why are you answering in a comment? $\endgroup$ – Arthur Nov 26 '19 at 17:04
  • $\begingroup$ @Arthur So the OP can learn and give an answer and receive feedback on it. $\endgroup$ – Gone Nov 26 '19 at 17:07
  • $\begingroup$ @BillDubuque Sure. But you can do that just as well, or even better, in an answer post. $\endgroup$ – Arthur Nov 26 '19 at 17:09
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    $\begingroup$ 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. $\endgroup$ – Gone Nov 26 '19 at 18:03
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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.

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