constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
There are multiple definitions of a Bézout ring:
A commutative ring is a Bézout ring if for every element and , there exists elements , called Bézout coefficients and called a common divisor, such that , and .
A commutative ring is a Bézout ring if it has functions , , such that for every element and , , and .
A commutative ring is a Bézout ring if it has functions , , , , such that for every element and , , and .
A commutative ring is a Bézout ring if every ideal with 2 generators is a principal ideal:
A commutative ring 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 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.
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
Last revised on August 19, 2024 at 15:05:07. See the history of this page for a list of all contributions to it.