Ideal

Relatively prime ideals

Let be a commutative ring. Two ideals are said to be relatively prime iff .¹ ring

Properties

1. Suppose are pairwise relatively prime. Then
2. Suppose are each relatively prime with . Then .
3. Suppose are distinct nonzero prime ideals in a 1-dimensional ring. Then for .

Proof of 1–2

For ^P1, it suffices to show the case for . For any ideals we already have . Since , it holds in particular that for some . Thus for any , we have , proving ^P1.

By the hypothesis of ^P2, for each there exists an such that . Then

and

hence , proving ^P2.

For ^P3 let . We show that

To see this, note that every element of is a sum of elements of the form where and . But such a term is itself a sum of terms containing either at least elements of or at least elements of , implying it is either an element of or .

Now ^P3 follows from this fact and the 1-dimensionality of , since we have , and by maximality of we have .

Results

• Chinese remainder theorem for rings

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Footnotes

1. 2022. Algebraic number theory course notes, §1.3.3, p. 25 ↩