Principal Ideal Domain is Bézout Domain
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Theorem
Let $D$ be a principal ideal domain.
Then $D$ is a Bézout domain.
Proof
Let $J_1, J_2$ be ideals of $D$.
From Sum of Ideals is Ideal, $J_1 + J_2$ is an ideal of $D$.
By definition of principal ideal domain, $J_1 + J_2$ is principal.
Hence $D$ is a Bézout domain.
$\blacksquare$