nLab Bézout ring

Contents

Context

Algebra

Constructivism, Realizability, Computability

Contents

Definition

There are multiple definitions of a Bézout ring:

In commutative rings

Definition

A commutative ring RR is a Bézout ring if for every element aRa \in R and bRb \in R, there exists elements xRx \in R, yRy \in R called Bézout coefficients and gRg \in R called a common divisor, such that ax+by=ga \cdot x + b \cdot y = g, g|ag \vert a and g|bg \vert b.

Definition

A commutative ring RR is a Bézout ring if it has functions β 0:R×RR\beta_0:R \times R \to R, β 1:R×RR\beta_1:R \times R \to R, γ:R×RR\gamma:R \times R \to R such that for every element aRa \in R and bRb \in R, aβ 0(a,b)+bβ 1(a,b)=γ(a,b)a \cdot \beta_0(a, b) + b \cdot \beta_1(a, b) = \gamma(a, b), γ(a,b)|a\gamma(a, b) \vert a and γ(a,b)|b\gamma(a, b) \vert b.

Definition

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

Definition

A commutative ring RR 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 aβ 0(a,b)+bβ 1(a,b)=gcd(a,b)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.

See also

References

Last revised on May 25, 2022 at 17:50:15. See the history of this page for a list of all contributions to it.