Euclidean Domains, PIDs & UFDs

Euclidean ⊆ PID ⊆ UFD.

The idea

Euclidean domains, principal ideal domains, and unique factorization domains classify rings by how much of the arithmetic of $\mathbb{Z}$ they retain.

In $\mathbb{Z}$, three facts hold at once: we can divide with a remainder smaller than the divisor, every ideal consists of the multiples of a single number, and every integer factors into primes in essentially one way. In a general integral domain each of these can fail, and each defines its own class of ring.

A Euclidean domain is an integral domain with a size function on its nonzero elements such that for any $a$ and any $b \neq 0$ there are $q$ and $r$ with $a = qb + r$, where $r$ is either $0$ or smaller than $b$. A principal ideal domain (PID) is an integral domain in which every ideal is generated by a single element, $I = (d)$. A unique factorization domain (UFD) is an integral domain in which every nonzero non-unit is a product of irreducibles, unique up to reordering and unit factors.

The three classes are nested: every Euclidean domain is a PID, every PID is a UFD, and neither implication reverses — $\mathbb{Z}[x]$, for instance, has unique factorization but is not a PID. Placing a ring in the hierarchy records exactly which of the three conveniences of $\mathbb{Z}$ it keeps.

Ways to work on it

• Walkthrough. The hierarchy and gcds in ℤ.
• Practice. Compute a gcd via the Euclidean algorithm.
• Hardest. Place a polynomial ring correctly in the hierarchy.

Not sure where to start? Take the ten-question placement test.