Over principal ideal domains In , the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences, and of a ring isomorphism. The statement in terms of remainders does not apply, in general, to principal ideal domains, as remainders are not defined in such rings. However, the two other versions make sense over a principal ideal domain : it suffices to replace "integer" by "element of the domain" and by . These two versions of the theorem are true in this context, because the proofs (except for the first existence proof), are based on Euclid's lemma and Bézout's identity, which are true over every principal domain.
Over principal ideal domains In , the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences and of a ring isomorphism. The statement in terms of remainders does not apply, in general, to principal ideal domains, as remainders are not defined in such rings. However the two other versions make sense over a principal ideal domain : it suffices to replace "integer" by "element of the domain" and by . These two versions of the theorem are true in this context, because the proofs (except for the first existence proof), are based on Euclid's lemma and Bézout's identity, which are true over every principal domain.