Corollary

— Let $(x_i)$ be an arbitrary family of elements of a principal ideal domain $A$ contained as a subring in an integral domain $B$, and let $d$ be a gcd of the family $(x_i)$ in $A$. Then the family $(x_i)$ has gcd's in $B$, and $d$ is one of them.

Indeed $d$ is a common divisor of the $x_i$ in $B$. On the other hand the relation $d = \sum a_i x_i$ shows that every common divisor of the $x_i$ in $B$ divides $d$.

An important application of this corollary is when $A = K[X]$ and $B = E[X]$, where $K$ is a field and $E$ is an extension of $K$ (IV, p. 12, Cor. 1).

The first assertion of Prop. 1 shows that the ordered group $\mathscr{P}^*$ is lattice ordered ($VI,$ p. 10). In particular every finite family of elements of $K$ admits an lcm. We can thus apply the results denoted (DIV) in $VI$, pp. 10 to 17 to principal ideal domains.

The following result is a consequence of the second assertion of Prop. I :