nLab Bézout ring

Redirected from "Bézout rings".
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 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, q 0:R×RRq_0:R \times R \to R, q 1:R×RRq_1: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), q 0(a,b)γ(a,b)=aq_0(a, b) \cdot \gamma(a, b) = a and q 1(a,b)γ(a,b)=bq_1(a, b) \cdot \gamma(a, b) = 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.

The third definition implies that Bézout rings are algebraic.

See also

References

Last revised on August 19, 2024 at 15:05:07. See the history of this page for a list of all contributions to it.