Divisors in commutative rings

There is a general notion of divisibility and in particular, that of a greatest common divisor (GCD) for any commutative ring \(R\):
1. \(a \mid b\) if \(b = ra\) for some \(r \in R\).
2. A GCD \(d\) of \(a\) and \(b\) must satisfy the following: (1) \(d \mid a\) and \(d \mid b\); (2) For any \(d'\) satisfying 1, \(d' \mid d\).

These statements can be expressed in terms of ideals:
1*. \(a \mid b\) if \((b) \subseteq (a)\).
2*. A GCD \(d\) of \(a\) and \(b\) must satisfy the following: (1) \((a,b) \subseteq (d)\); (2) For any \(d'\) satisfying 1, \((d) \subseteq (d')\).
(Note that \((a,b)\) is the smallest ideal containing \((a)\) and \((b)\), so \((a),(b) \subseteq (d)\) automatically implies \((a,b) \subseteq (d)\).)

Based on the definition, we see that \((d)\) must be the unique smallest principal ideal containing \((a,b)\). This rules out situations where there are at least 2 minimal principal ideals containing \((a,b)\). Also note that there can exist multiple GCDs, provided they generate the same principal ideal. To take an extreme example, every real number is a GCD of every pair of real numbers (this is true for fields in general).

Generally, we do not require \((a,b) = (d)\). For example, in the polynomial ring \(\mathbb{Z}[x]\), \((2,x)\) is a maximal non-principal ideal (because the associated quotient is a field). Thus 1 is the GCD of 2 and \(x\) even though \((2,x) \subset (1)\).

On the other hand, there is a hierarchy of rings satisfying \((a,b) = (d)\) for all \(a,b \in R\):
Euclidean domains \(\subset\) PIDs \(\subset\) Bezout domains \(\subset\) Commutative rings

Bezout domains are defined precisely to satisfy this condition. However, one must note that not all Bezout domains are principal ideal domains. For instance, they can have infinitely generated non-principal ideals.

PIDs satisfy this condition trivially; in fact all PIDs are Bezout domains. In particular, Euclidean domains enable one to algorithmically compute \((d)\) and express \(d\) as an \(R\)-linear combination of \(a\) and \(b\), via the Euclidean algorithm.