CRT in General

Recall that all rings in this book are commutative with unity.

Definition 5.1.1 (Coprime)   Ideals and are coprime if .

If  and  are nonzero ideals in the ring of integers of a number field, then they are coprime precisely when the prime ideals that appear in their two (unique) factorizations are disjoint.

Lemma 5.1.2   If and are coprime ideals in a ring , then .

Proof. Choose and such that . If then

so . The other inclusion is obvious by definition of ideal.

Lemma 5.1.3   Suppose are pairwise coprime ideals. Then is coprime to the product .

Proof. It suffices to prove the lemma in the case , since the general case then follows from induction. By assumption, there are and such

and

Multiplying these two relations yields

The first three terms are in and the last term is in (by Lemma 5.1.2), so is coprime to .

Next we prove the general Chinese Remainder Theorem. We will apply this result with in the rest of this chapter.

Theorem 5.1.4 (Chinese Remainder Theorem)   Suppose are nonzero ideals of a ring  such and are coprime for any . Then the natural homomorphism induces an isomorphism

Thus given any , for , there exists some such that for ; moreover,  is unique modulo .

Proof. Let be the natural map induced by reduction modulo the . An inductive application of Lemma 5.1.2 implies that the kernel of  is equal to , so the map  of the theorem is injective.

Each projection is surjective, so to prove that is surjective, it suffices to show that is in the image of , and similarly for the other factors. By Lemma 5.1.3, is coprime to , so there exists and such that . Then maps to  in and to 0 in , hence to 0 in for each , since .

William Stein 2012-09-24