Recall that all rings in this book are commutative with unity.
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.
If and are coprime ideals in a ring , then
The other inclusion is obvious by definition of ideal.
are pairwise coprime ideals.
Then is coprime to the product
It suffices to prove the lemma in the case
, since the
general case then follows from induction.
By assumption, there
Multiplying these two relations yields
The first three terms are in
and the last term is in
(by Lemma 5.1.2
is coprime to
Next we prove the general Chinese Remainder Theorem.
We will apply this result with in the rest of this chapter.