# Ring in which Quotient and Remainder not Unique

• Bashyboy
In summary, the author is trying to solve a problem involving factorization and exceptions, but is having difficulty because he has not been introduced to matrix rings. He suggests looking at polynomial rings or the ring of square roots of negative integers.
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)##.

## 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.

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]##.

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.

Bashyboy said:
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}]##.

Unfortunately these haven't been introduced either.

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.

## 1. What is a ring in which quotient and remainder are not unique?

A ring is a mathematical structure consisting of a set of elements and two binary operations: addition and multiplication. In a ring where the quotient and remainder are not unique, there may be multiple ways to divide one element by another and obtain a quotient and remainder.

## 2. How is this different from a traditional ring?

In a traditional ring, the quotient and remainder are unique for any division operation. This means that given any two elements in the ring, there is only one way to divide one by the other and obtain a quotient and remainder.

## 3. What causes the quotient and remainder to not be unique?

The quotient and remainder may not be unique in a ring if the ring is not an integral domain. This means that there are elements in the ring that do not have unique multiples of other elements in the ring. This can happen if there are zero divisors in the ring, meaning there are non-zero elements that multiply to give zero.

## 4. What implications does this have for mathematical calculations?

In a ring where the quotient and remainder are not unique, there may be different ways to simplify or manipulate mathematical expressions involving division. This can lead to different results and may require additional steps to ensure accuracy in calculations.

## 5. Can a ring with non-unique quotient and remainder still be useful in mathematics?

Yes, rings with non-unique quotient and remainder can still have important applications in mathematics. For example, they can be used in cryptography and coding theory. However, it is important to understand their properties and limitations in order to use them effectively.

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