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$\color{Green}{Background:}$

$\textbf{Definition:}$ An integral domain $R$ is a Euclidean domain if there is a function $\delta$ from the nonzero elements of $R$ to the nonnegative integers with these properties;

$(i)$ If $a$ and $b$ are nonzero elements of $R,$ then $\delta(a)\leq \delta(ab).$

$(ii)$ If $a,b\in R$ and $b\neq 0_R,$ then there exist $q,r\in R$ such that $a=bq+r$ and either $r=0_R$ or $\delta(r) < \delta(b).$

$\textbf{Exercise 1:}$ Let $R$ be a Euclidean ring with Euclidean norm $\delta$. Let $a,b\in R\setminus\{0\}$ and let $q,r\in R$ such that $a=bq+r$ with $r=0$ or $\delta(r)<\delta(b)$.
Prove that $r$ and $q$ are unique if and only if $\delta(a+b)\le\max\{\delta(a),\delta(b)\}$.

Variant wording:

$\textbf{Exercise 1a:}$ Let $R$ be a Eucliean domain such that $\delta(a+b)\leq \text{max}(\delta(a),\delta(b))$ for all nonzero $a,b\in R.$ Prove that $q$ and $r$ in the definition of Euclidean domain are unique.

Example and non-example to $\textbf{Exercise 1:}$ of $\delta$ functions in the case of the ring $\mathbb{Z}:$

The ring $\mathbb{Z}$ is Euclidean with respect to the norm $\delta(z)=|z|$ for $z\in \mathbb{Z}.$ For non-zero $a,b \in \mathbb{Z}$ we have $1\leq |b|$ and so $|a|\leq |a||b|=|ab|.$ Thus $\delta(a)\leq \delta(ab).$

It is possible to have more than one norm function on a Euclidean domain. We shall show that $\mathbb{Z}$ is also Euclidean with respect to the given by $\delta(z)=z^2$ for all $z\in \mathbb{Z}.$ Given $a,b\in \mathbb{Z}$ with $b\neq 0,$ we know that $a=bq+r$ for some $q,r\in \mathbb{Z}$ with $0\leq r < |b|.$ Clearly then we have either $r=0$ or $r^2< b^2,$ i.e. either $r=0$ or $\delta(r)<\delta(b).$ Also $\delta(a)=a^2\leq a^2b^2=(ab)^2=\delta(ab).$ Thus $\mathbb{Z}$ is Euclidean with respect to $\delta.$ Here the quotient and the remainder are not unique. Take $a=3$ and $b=2.$ We have $a=bq+r$ where either $q=r=1$ or $q=2$ and $r=-1.$ In both cases $\delta(r)=1<4=\delta(b).$

$\color{Red}{Questions:}$

For the Exercise 1/1a above, I was trying to find examples of Euclidean domains with a $\delta$ function that either satisfy $\delta(a+b)\le\max\{\delta(a),\delta(b)\}$ or breaks it. I did find it for the case of $\mathbb{Z}$ with $\delta(z)=|z|$ and $\delta(z)=z^2.$ With $a=3, b=1$ for the case of $\delta(z)=|z|,$ we have $\delta(2)=|2|,$ $\delta(3)=|3|,$ $\delta(2+3)=|5|,$ but $\delta(2+3)>\max\{\delta(2),\delta(3)\}.$ Similarily for the case of $\delta(z)=z^2,$ with $\delta(2)=4,$ $\delta(3)=9,$ $\delta(2+3)=25.$

But the division algorithm with uniquess of quotient and remainder holds for the ring $\mathbb{Z}.$ My understanding is that one can check whether the division algorithm have unique quotient and remainder in the case of Euclidean ring by checking whether such ring's $\delta$ function satisfies the criteria of the exercise. Also, are there examples of Euclidean rings that one needs to judiciously choose a $\delta$ function to satisfies the above exercise, so that even though ifeven though alternate $\delta$ function don't satisfy the above exercise's criteria, but because it has been shown that it is so for a $\delta$ function, then uniquenes of quotient and remainder holds regardless if other $\delta$ functions for the same Eucldiean ring fails the criteria of the above exercise.

Thank you in advance

Seth
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  • Are you sure there's uniqueness for $\Bbb{Z}$? If $a=17, b=5$ you have $17 = 3 * 5 + 2$ and $17 = 4 * 5 - 3$. – Chad K Dec 20 '23 at 21:10
  • @ChadK hm...maybe I am misunderstanding what the question means by uniqueness. If it is not too much trouble can I ask you to show an example to illustrate what the question mean by uniqueness and examples where it fails in the context of the exercise's criteria. – Seth Dec 20 '23 at 21:32
  • The uniqueness depends on $\delta$, because a solution requires $\delta(r)<\delta(b)$, and what's a solution for one $\delta$ may not be a solution for another $\delta$ on the same ring. You showed yourself that $|z|$ and $z^2$ fail the test for $\Bbb{Z}$, and my example shows there's no uniqueness for either of these norms. OTOH, for the polynomial ring $K[X]$ over a field, the degree norm satisfies the test, and I believe there's uniqueness there. – Chad K Dec 20 '23 at 21:40
  • @ChadK can I conclude from your comment that if I want Euclidean rings with $\delta$ function that satisfies the criteria of the exercise for uniqueness, I should look to Euclidean rings that are also a field? – Seth Dec 20 '23 at 22:02
  • No. $K$ is a field. $K[X]$ the polynomial ring in one variable over $K$ is not a field. It's a Euclidean domain with $\delta(f)=\deg f$. – Chad K Dec 20 '23 at 22:20
  • @ChadK are there other examples besides polynomial rings? I am trying to have an intuition for the exercise. – Seth Dec 20 '23 at 22:32
  • @Seth did you get to the bottom of this? In my book, the proof to the division algorithm in $\mathbb{Z}$ is "use $\delta(a)=|a|$" with the theorem you posted ($\delta(a+b)\le max(\delta(a),\delta(b))$). I don't understand how this works – hirdajarzu Mar 26 '24 at 13:10
  • @hirdajarzu sorry you wrote a book? – Seth Mar 26 '24 at 18:40
  • @Seth no; in the book I use to study – hirdajarzu Mar 29 '24 at 14:09

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