nLab Bézout ring

Definition

There are multiple definitions of a Bézout ring:

In commutative rings

Definition

A commutative ring $R$ is a Bézout ring if for every element $a \in R$ and $b \in R$, there exists elements $x \in R$, $y \in R$ called Bézout coefficients and $g \in R$ called a common divisor, such that $a \cdot x + b \cdot y = g$, $g \vert a$ and $g \vert b$.

Definition

A commutative ring $R$ is a Bézout ring if it has functions $\beta_0:R \times R \to R$, $\beta_1:R \times R \to R$, $\gamma:R \times R \to R$ such that for every element $a \in R$ and $b \in R$, $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gamma(a, b)$, $\gamma(a, b) \vert a$ and $\gamma(a, b) \vert b$.

Definition

A commutative ring $R$ is a Bézout ring if it has functions $\beta_0:R \times R \to R$, $\beta_1:R \times R \to R$, $\gamma:R \times R \to R$, $q_0:R \times R \to R$, $q_1:R \times R \to R$ such that for every element $a \in R$ and $b \in R$, $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gamma(a, b)$, $q_0(a, b) \cdot \gamma(a, b) = a$ and $q_1(a, b) \cdot \gamma(a, b) = b$.

Definition

A commutative ring $R$ is a Bézout ring if every ideal with 2 generators is a principal ideal:

Definition

A commutative ring $R$ is a Bézout ring if every finitely generated ideal is a principal ideal.

If the commutative ring is a GCD ring and the common divisor is the greatest common divisor, then the Bézout ring condition $a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gcd(a, b)$ is called the Bézout identity. There might be multiple Bézout coefficient functions for each Bézout ring.

The third definition implies that Bézout rings are algebraic.

See also

• commutative ring

• GCD ring

• Bézout domain

• Euclidean ring

References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)

• Wikipedia, Bézout’s identity