Bézout domain

Remarks

Equivalent form: for all $a, b \in R$ the ideal $(a, b)$ is principal; the general case follows by induction on the number of generators. Then $(a, b) = (d)$ for a gcd $d$ of $a$ and $b$, and $d = ax + by$ for some $x, y \in R$. Example: every principal ideal domain, such as $\Z$ or $\Q[x]$, is a Bézout domain. Non-example: in $\Z[x]$ the ideal $(2, x)$ is not principal. A generator would divide $2$, so it would be $\pm 1$ or $\pm 2$. But $(2, x) \ne \Z[x]$, because its elements have even constant term, and $2$ does not divide $x$. A Bézout domain is a GCD domain; a Noetherian Bézout domain is a principal ideal domain.