Proposition 2

— Let $a, b, d, m$ and $p$ be elements of the field of fractions $K$ of a principal ideal domain $A$. a) « $d$ is a gcd of $a$ and $b$ » is equivalent to « $(d) = (a) + (b)$ ». b) « $m$ is an lcm of $a$ and $b$ » is equivalent to « $m = (a) \cap (b)$ ». c) « $p$ is an irreducible element of $A$ » is equivalent to « $(p)$ is a nonzero maximal ideal of $A$ » and to « $(p)$ is a nonzero prime ideal of $A$ ».

We have already proved a) (Prop. 1). Since the common multiples of $a$ and $b$ are the elements of $(a) \cap (b)$, and since, by hypothesis, the ideal $(a) \cap (b)$ is principal, say $(a) \cap (b) = (m)$, it follows that $m$ is an Icm of $a$ and $b$, which proves b). Finally, to say that $p \neq 0$ is an irreducible element of $A$ means by definition (VI, p. 17) that $(p)$ is a maximal element of the family of principal ideals $\neq A$ of $A$, ordered by inclusion; since $A$ has no ideals other than the principal ideals, this means that $(p)$ is a maximal ideal of $A$, whence c), by the remark in VI, p. 17.

In a principal ideal domain $A$, the sum (resp. the intersection) of a finite number of ideals is also called the gcd (resp. lcm) of these ideals.