Integral domain

An integral domain is a nonzero commutative ring with identity and without zero divisors.

Multiplication by a nonzero element of an integral domain is injective. If the ring is finite, that map is surjective, so its image contains one and the multiplier has a multiplicative inverse. Hence every nonzero element is invertible and the ring is a field.

A nonzero nonunit $p$ in an integral domain is prime when $p\mid ab$ implies $p\mid a$ or $p\mid b$.

A nonzero nonunit $p$ in an integral domain is irreducible when every factorization $p=ab$ has at least one unit factor.

An atomic domain is an integral domain in which every nonzero nonunit is a finite product of irreducible elements. Every unique factorization domain is atomic, but atomicity alone does not imply uniqueness.

A greatest-common-divisor domain is an integral domain in which every two nonzero elements have a greatest common divisor, defined up to multiplication by a unit.

If $\gcd (a,b)=1$ and $a\mid bc$ in a greatest-common-divisor domain, then $a\mid c$. Indeed, the least common multiple of $a$ and $b$ is associate to $ab$ and divides every common multiple; applying this to $bc$ and cancelling $b$ proves the claim.

The fraction field of an integral domain $R$ consists of fractions $a/b$ with $a,b\in R$ and $b\ne 0$, modulo the usual equivalence relation.

A principal ideal domain is an integral domain whose every ideal has one generator.

The ascending chain condition supplies existence of factorizations into irreducible elements. The Bezout identity makes every irreducible element prime, which supplies uniqueness up to order and multiplication by units.

Every submodule of a finite-rank free module over a principal ideal domain is free. Induction on rank splits off a generator of the image under one coordinate projection and applies the result to the kernel.

A Euclidean domain admits division with remainder measured by a decreasing Euclidean function.

A Euclidean function on an integral domain $R$ assigns a nonnegative integer to every nonzero element so that, for $a,b\in R$ with $b\ne 0$, there are $q,r\in R$ satisfying $a=bq+r$ and either $r=0$ or $d(r)<d(b)$.

A Gaussian integer has the form $a+bi$ with $a,b\in \mathbb{Z}$. Its multiplicative norm is $N(a+bi)=a^2+b^2$, which makes $\mathbb{Z}[i]$ a Euclidean domain.

If $\alpha =a+bi$ is a Gaussian integer and $P$ is a nonzero prime ideal of $R=\mathbb{Z}[\alpha ]$, then $R/P$ is a finite field. Complex conjugation preserves $R$, so a nonzero $\beta \in P$ gives the positive integer $\beta \bar{\beta }\in P$. Primality supplies a rational prime $p\in P$. Since $\alpha$ satisfies the monic polynomial $t^2-2at+a^2+b^2$, the ring $R/P$ is a quotient of ${\mathbb{F}}_p[t]/(\bar{F})$ and has at most $p^2$ elements. It is a finite integral domain, hence a field. The proof does not require $R$ to be a principal ideal domain.

For a primitive cube root of unity $\omega$, the Eisenstein integers form the triangular latticeTheir multiplicative norm is $N(a+b\omega )=a^2-ab+b^2$.

The Eisenstein-integer norm is $N(z)=z\overline{z}$. It is multiplicative and takes $a+b\omega$ to $a^2-ab+b^2$.

Every complex number is within squared distance at most $3/4$ of an Eisenstein integer: write it as $x+y\omega$ and round $x$ and $y$ to integers. Therefore, for nonzero $\beta$ and arbitrary $\alpha$, a nearest lattice point $q$ to $\alpha /\beta$ givesThe norm makes the Eisenstein integers a Euclidean domain.

A Bézout domain is an integral domain in which every finitely generated ideal is principal. Equivalently, every pair $a,b$ has a greatest common divisor $d$ satisfying a Bézout identity $d=ra+sb$. A Noetherian Bézout domain is a principal ideal domain because all its ideals are finitely generated.