§2T. Euclidean Domains, Principal Ideal Domains, and Unique Factorization Domains

## Euclidean domains, principal ideal domains, and unique factorization domains

$R$ is a Euclidean domain $⇒R$ is a PID.

Definition.

• Let $ℕ=\left\{0,1,2,\dots \right\}$ be the set of nonnegative integers. A Euclidean domain is an integral domain $R$ with a function $σ:R∖0→ℕ,$ 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 $⇒R$ 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
1. $p\ne 0$,
2. $p$ is not a unit,
3. $p$ 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

1. $p$ is a unit $⇔\left(p\right)=R$.
2. $p$ divides $q⇔\left(q\right)\subseteq \left(p\right)$.
3. $p$ is a proper divisor of $q⇔\left(q\right)⊊\left(p\right)⊊R$.
4. $p$ is an associate of $q⇔\left(q\right)=\left(p\right)$.
5. $p$ is irreducible $⇔$
1. $\left(p\right)\ne 0$,
2. $\left(p\right)\ne R$,
3. 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 $0⊊ a1⊊ a2⊊⋯$

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
1. $d$ divides ${a}_{i}$ for all $i=0,1,\dots n$
2. 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

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

## References

[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)