Last updates: 08 March 2011

$R$** is a Euclidean domain **$\Rightarrow R$ **is a PID.**

**Definition.**

- Let $\mathbb{N}=\left\{0,1,2,\dots \right\}$ be the set of nonnegative integers. A
**Euclidean domain**is an integral domain $R$ with a function $$\sigma :R\setminus \left\{0\right\}\to \mathbb{N},$$ called a**size function**, such that if $a,b\in R$ and $a\ne 0$, then there exist $q,r\in R$ such that $$b=aq+r,$$ where either $r=0$ or $\sigma \left(r\right)<\sigma \left(a\right)$. - Let $R$ be a commutative ring. A
**principal ideal domain**is an ideal generated by a single element. - A
**principal ideal domain**(or**PID**) is an integral domain for which every ideal is principal.

A Euclidean domain is a principal ideal domain.

$R$** is a PID **$\Rightarrow R$** is a UFD.**

**Definition.** Let $R$ be an integral domain.

- A
**unit**is an element $a\in R$ such that there exists an element $b\in R$ such that $ab=1$. - Let $p,q\in R$. The element $p$
**divides**$q$ if $q=ap$ for some $a\in R$. - Let $p,q\in R$. The element $p$ is a
**proper divisor**of $q$ if $q=ap$ for some $a\in R$ and neither $a$ nor $p$ is a unit. - Let $p,q\in R$. The elements $p$ and $q$ are
**associates**if $p=aq$ for some unit $a\in R$. - An element $p\in R$ is
**irreducible**if- $p\ne 0$,
- $p$ is not a unit,
- $p$ has no proper divisor.

The following proposition shows that every property of divisors can be written in terms of containments of ideals and vice versa.

Let $p,q\in R$ and $\left(p\right)$ and $\left(q\right)$ denote the ideals generated by the elements $p$ and $q$ respectively. Then

- $p$ is a unit $\iff \left(p\right)=R$.
- $p$ divides $q\iff \left(q\right)\subseteq \left(p\right)$.
- $p$ is a proper divisor of $q\iff \left(q\right)\u228a\left(p\right)\u228aR$.
- $p$ is an associate of $q\iff \left(q\right)=\left(p\right)$.
- $p$ is irreducible $\iff $
- $\left(p\right)\ne 0$,
- $\left(p\right)\ne R$,
- If $q\in R$ and $\left(q\right)\supseteq \left(p\right)$ then either $\left(q\right)=\left(p\right)$ or $\left(q\right)=R$.

**Definition.**

- A
**unique factorization domain**(or**UFD**) is an integral domain such that - If $x\in R$ then $x={p}_{1}{p}_{2}\cdots {p}_{n}$ for some ${p}_{1},\dots ,{p}_{n}\in R$ which are all irreducible.
- If $x\in R$ and $x={p}_{1}{p}_{2}\cdots {p}_{n}=u{q}_{1}{q}_{2}\cdots {q}_{m}$ where $u\in R$ is a unit and ${p}_{1}{p}_{2}\cdots {p}_{n},{q}_{1}{q}_{2}\cdots {q}_{m}\in R$ are irreducible, then $m=n$ and there exists a permutation $\sigma :\left\{1,2,\dots n\right\}\to \left\{1,2,\dots n\right\}$ such that for each $1\le i\le n$, ${q}_{i}={u}_{i}{p}_{\sigma \left(i\right)},$ for some unit ${u}_{i}\in R$.

A principal ideal domain is a unique factorization domain.

The proof of Theorem 1.3 will require the following lemmas.

If $R$ is a principal ideal domain and $p\in R$ is an irreducible element of $R$ then $\left(p\right)$ is a prime ideal.

Let $R$ be a principal ideal domain. There does **not** exist an infinite sequence of elements
${a}_{1},{a}_{2},\dots \in R$
such that
$$\left(0\right)\u228a\left({a}_{1}\right)\u228a\left({a}_{2}\right)\u228a\cdots $$

**Greatest common divisors**

**Definition.**

- Let $R$ be a unique factorization domain. Let
${a}_{0},{a}_{1},\dots ,{a}_{n}\in R.$
A
**greatest common divisor**, $gcd\left({a}_{0},{a}_{1},\dots ,{a}_{n}\right)$ is an element $d\in R$ such that- $d$ divides ${a}_{i}$ for all $i=0,1,\dots n$
- If $d\text{'}$ divides ${a}_{i}$ for all $i=0,1,\dots n$, then $d\text{'}$ divides $d$.

Let $R$ be a unique factorization domain and let ${a}_{0},{a}_{1},\dots ,{a}_{n}\in R.$ Then

- $gcd\left({a}_{0},{a}_{1},\dots ,{a}_{n}\right)$ exists.
- $gcd\left({a}_{0},{a}_{1},\dots ,{a}_{n}\right)$ is unique up to multiplication by a unit.

[CM]
H. S. M. Coxeter and W. O. J. Moser, *Generators and relations for discrete groups*,
Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980.
MR0562913 (81a:20001)

[GW1]
F. Goodman and H. Wenzl,
*The Temperly-Lieb algebra at roots of unity*, Pacific J. Math. 161 (1993), 307-334.
MR1242201 (95c:16020)