Definition 5

— In an ordered group two elements x and y are called coprime if $\inf(x, y) = 0$.

In some cases it is natural to define two elements to be coprime if $\inf(|x|, |y|) = 0$ (cf. INT, II, § 1) or to introduce the corresponding terminology in divisibility theory. We shall not do so here.

Two coprime elements are necessarily positive. The positive and negative parts $x^+$ and $x^-$ of x are coprime (VI, p. 12, Prop. 9, a)). The elements $x_i$ of a family $(x_i)_{i \in I}$ are said to be setwise coprime if $\inf_{i \in I} x_i = 0$; if the $x_i$ are all $\geq 0$ then it is sufficient for there to exist a finite subset J of I such that the corresponding elements are setwise coprime. The elements of a family $(x_i)$ are said to be pairwise coprime if $\inf(x_i, x_j) = 0$ for every pair $(\ell, \kappa)$ of distinct indices.

The $x_i$ can be setwise coprime without being pairwise coprime.

If x and y are coprime, we also say that x is coprime to y, or that y is coprime to X

(DIV) Two elements x and y of K are said to be coprime if the principal ideals (x) and (y) are nonzero and coprime in $\mathscr{P}^*$; this amounts to saying that 1 is a gcd of x and y, and implies that x and y belong to A. For example the numerator and denominator of a reduced fraction are coprime. The notions of pairwise and setwise coprime elements are defined similarly.

(DIV) When x and y are coprime, they are often said to be « prime to one another »; it is convenient to avoid this terminology, which can lead to confusion with the notion of prime numbers (I, p. 50, Def. 16).