In algebra, one of the central structures is that of the ring — specifically here, we focus on unital, commutative rings (i.e., those with commutative multiplication and a multiplicative identity) that admit no zero-divisors. These are known as integral domains, and there are a few major ones that we will focus on.

Euclidean Domains

From grade school, we know that we can divide two integers together with remainder — i.e., if $a,b\in \mathbb{Z}$, then there are unique $q$ and $r$ such that $ a = qb + r,$ where $r = 0$ or $|r| < b$. Then, in high school, we learn that if we have two polynomials (with real coefficients), $a(x)$ and $b(x)$, then we may divide $a(x)$ by $b(x)$ to find an expression of the form $a(x) = q(x)b(x) + r(x)$, where $r(x) = 0$ or $\deg(r) < \deg(b)$.

These are the two archetypal examples of a Euclidean domain, which admits a much more formal definition as follows.

We say $R$ is a Euclidean domain if $R$ admits a norm $N\colon R\setminus {0}\rightarrow \mathbb{N}$ such that any two elements $a,b\in R\setminus {0}$ admit division with remainder. That is, we may write $a = qb + r$, where $r = 0$ or $N(r) < N(b)$.

Division with remainder is a very powerful tool, and we will see soon enough that this allows Euclidean domains to admit many nice properties. One of those is that Euclidean domains are Principal Ideal Domains, which we discuss next.