Ring in which Quotient and Remainder not Unique

[Bashyboy]

Homework Statement

Give an example of a commutative ring $R$ and $f(x), g(x) R[x]$ with $f$ monic such that the remainder after dividing $g$ by $f$ is not unique; that is, there are $q,q',r,r' \in R[x]$ with $qf + r = g = q' f + r'$ and $\deg (r)$ and $\deg (r')$ are both strictly less than $\deg(f)$.

Homework Equations

The Attempt at a Solution

Okay. I have been thinking about this problem for a rather long time ---I've been foiling for hours (perhaps an exaggeration). I have tried several pairs of polynomials when the base ring is $\Bbb{Z}_4$, $\Bbb{Z}_6$, and $\Bbb{Z}_8$. My strategy was to solve for $q$ when $r=0$; and then find a $q'$ given some nonzero $r'$. I set up with equations with five variables so that I could find a nonzero remainder with $\deg (r) < \deg (f)$, but I could find no values for these five variables. I could use a hint on finding a ring $R$ and polynomials $f(x)$ and $g(x)$. But please don't just hand them to me; I would like to figure out this problem on my own as much as possible.

 

[fresh_42]

I don't know an example either, but when it comes to factorization and exceptions, the matrix rings are a good address to look for examples. This way you avoid all the stuff which comes in with fields and their extensions. Or you could look at $R[x,y]=R[y][x]$.

 

[Bashyboy]

Well, since matrix rings haven't been introduced yet, I suspect that the author of the book I am using has something more elementary in mind.

 

[fresh_42]

Well, since matrix rings haven't been introduced yet, I suspect that the author of the book I am using has something more elementary in mind.

Try $\mathbb{Z}[\sqrt{-3}]$ or $\mathbb{Q}[-\sqrt{5}]$.

 

[Bashyboy]

Unfortunately these haven't been introduced either.

 

[fresh_42]

But polynomial rings have been introduced? So what about my other example $R[y]$ as ring and polynomial ring $R[y][x]\,$? But I haven't thought about whether the fact that it is no principal ideal domain is already sufficient to supply a counterexample.