Abs. Alg. by Dummit and Foote - Chapter 8 Notes

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Chapter 8 - Euclidean, Principal Ideal, and Unique Factorization Domains

Section 8.1 - Euclidean Domains

Definition: Euclidean Domain

Proposition 1

Every ideal in a Euclidean Domain is principal. In particular, for any nonzero ideal $I$, we have that $I=(d)$ for some nonzero element $d\in I$ with minimal norm.

Proof

Let $I$ be a nonzero ideal in a Euclidean Domain $R$. Choose nonzero $d\in I$ such that $d$ has minimal norm (which must exist by the well ordering of $\mathbb{Z}$). Clearly, $(d)\subseteq I$ since $d\in I$, so it remains to show that $I\subseteq (d)$. Take $a\in I$, then by the existence of a division algorithm in $R$, we have that for $a,d$ there exist $q,r\in R$ such that,

$$a=qd+r\text{ with }r=0\text{ or }N(r)<N(d).$$

Since $a,d\in I$, we have from the closure of $I$ that $q,r\in I$. Since $d$ was chosen to be the element with minimal nonzero norm in $I$, we know that $r=0$ (otherwise $N(r)<N(d)$ wouldn’t hold), so we have that $a=qd$ an element of $(d)$, so $I\subseteq (d)$. Therefore, we have that $I=(d)$. $\square$

It is by proposition 1 that we can show that some integral domains are not Euclidean Domains by showing that they have non-principal ideals.

Definition: Greatest Common Divisor (in a Ring)

Just like in $\mathbb{Z}$, the idea of a Euclidean Domain allows us to construct a generalized version of greatest common divisor.

Note that $b\mid a$ iff $a\in (b)$ iff $(a)\subseteq (b)$. This ends up giving us that the greatest common divisor (if it exists) of $a,b\in R$ generates the unique smallest principal ideal containing $(a)$ and $(b)$.

It is important to note that not all rings have greatest common divisors.