Lemma 1 ("Euclid's Lemma")

Let $x, y$ be two strongly relatively prime elements of $R$; if $z \in R$ is such that $x$ divides $yz$, then $x$ divides $z$.

If $1 = ax + by$, then $z = x(az) + (yz)b$.

If $x$ and $y$ are strongly relatively prime in $R$, then

(Chapter II, § 1, no. 2, Proposition 5); if $R$ is an integral domain, two strongly relatively prime elements then have an lcm equal to their product (Algebra, Chapter VI, § 1, no. 8) and are therefore relatively prime in the sense of Algebra, Chapter VI, § 1, no. 12. Conversely, if $R$ is a principal ideal domain, two relatively prime elements are also strongly relatively prime, as follows from Bezout's identity (Algebra, Chapter VII, § 1, no. 2, Theorem 1).

For polynomial rings there is the following result: