# Definition:Bézout Domain

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## Definition

### Definition 1

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

### Definition 2

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.

## Historical Note

The definition of an integral domain, and even that of a ring, was not formulated until over a century after the death of Étienne Bézout.

However, a Bézout domain bears his name because in such an algebraic structure, each pair of elements satisfies an algebraic formulation of Bézout's Identity.