# Definition:Bézout Domain

## Definition

A **Bézout domain** is an integral domain in which the sum of two principal ideals is again principal.

Equivalently, a **Bézout domain** is an integral domain in which every finitely generated ideal is principal.

## Source of Name

This entry was named for Étienne Bézout.

Although even the definition of a ring, let alone that of an integral domain, was not formulated until over a century after his death, a **Bézout Domain** bears his name because in such a structure, each pair of elements satisfies an algebraic formulation of Bézout's Identity.