pairwise comaximal ideals property

Proposition 1.

Let R be a commutative ring with unity. For every pairwise comaximal ideals I1,I2,,In, the following holds:

I1I2In=I1I2In. (1)

We prove by inductionMathworldPlanetmath on n. For n=2, I1+I2=R implies:

I1I2=R(I1I2)=(I1+I2)(I1I2)I1I2. (2)

The converseMathworldPlanetmath inclusion is trivial. Assume now that the equality holds for n2: J:=I1I2In=I1I2In. Since In+1+Ij=R, for every jn+1, there exist the elements ajIj and bjIn+1 such that aj+bj=1. The productPlanetmathPlanetmath c:=j=1naj=j=1n(1-bj)1+In+1. Also cJ, then 1J+In+1 or J+In+1=R.
Applying the case 2, the induction step is satisfied:

I1I2In+1=JIn+1=JIn+1=I1I2InIn+1. (3)

Title pairwise comaximal ideals property
Canonical name PairwiseComaximalIdealsProperty
Date of creation 2013-03-22 16:53:34
Last modified on 2013-03-22 16:53:34
Owner polarbear (3475)
Last modified by polarbear (3475)
Numerical id 9
Author polarbear (3475)
Entry type Result
Classification msc 16D25