divisibility by product


Let R be a Bézout ring, i.e. a commutative ring with non-zero unity where every finitely generatedMathworldPlanetmathPlanetmathPlanetmath ideal is a principal idealMathworldPlanetmathPlanetmath. If a,b,c are three elements of R such that a and b divide c and  gcd(a,b)=1,  then also ab divides c.

Proof. The divisibility assumptions that  c=aa1=bb1  where a1 and b1 are some elements of R.  Because R is a Bézout ring, there exist such elements x and y of R that  gcd(a,b)=1=xa+yb. This implies the equationa1=xaa1+yba1=xbb1+yba1  which shows that a1 is divisible by b, i.e.  a1=bb2,  b2R. Consequently,  c=aa1=abb2,  or  abc  Q.E.D.

Note 1. The theorem may by induction be generalized for several factors (http://planetmath.org/Divisibility) of c.

Note 2. The theorem holds e.g. in all Bézout domains, especially in principal ideal domainsMathworldPlanetmath, such as and polynomial ringsMathworldPlanetmath over a field.

Title divisibility by product
Canonical name DivisibilityByProduct
Date of creation 2013-03-22 14:50:37
Last modified on 2013-03-22 14:50:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Theorem
Classification msc 11A51
Classification msc 13A05
Related topic BezoutDomain
Related topic ProductDivisibleButFactorCoprime
Related topic CorollaryOfBezoutsLemma
Defines Bézout ring