well-definedness of product of finitely generated ideals

Ler R be of a commutative ring with nonzero unity.  If

𝔞=(a1,,am)=(α1,,αμ) (1)


𝔟=(b1,,bn)=(β1,,βν) (2)

are two finitely generatedMathworldPlanetmathPlanetmathPlanetmath ideals of R, both with two , then the ideals




are equal.

Proof.  By (1) and (2), for every i,j, there are elements rik,sjl of R such that

ai=ri1α1++riμαμ,bj=sj1β1++sjνβν. (3)

Multiplying the equations (3) we see that


whence the generatorsPlanetmathPlanetmathPlanetmath aibj of 𝔠 belong to 𝔡 and consecuently  𝔠𝔡.  The reverse containment is seen similarly.

Title well-definedness of product of finitely generated ideals
Canonical name WelldefinednessOfProductOfFinitelyGeneratedIdeals
Date of creation 2013-03-22 19:12:56
Last modified on 2013-03-22 19:12:56
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 16D25
Related topic WellDefined
Related topic ProductOfIdeals
Related topic ProductOfFinitelyGeneratedIdeals