Generalization to arbitrary rings The Chinese remainder theorem can be generalized to any ring, by using coprime ideals (also called comaximal ideals). Two ideals and are coprime if there are elements and such that This relation plays the role of Bézout's identity in the proofs related to this generalization, which otherwise are very similar. The generalization may be stated as follows.
generAlizaTioN To arbiTrary rings The chInESe rEMaIndeR thEorem cAn bE genERalizeD to aNy riNG by uSiNG CoPrime Ideals (aLSO calleD COMaximAl ideals). Two IdeaLS and ArE CopRime iF theRE ARe eLements aND sucH THAt this rElaTion Plays the rOle of bézOUt'S identity in THe prooFs RElateD to THis GenErAlizaTioN which othErWIse are veRY sIMilAr. ThE generaLizaTion may be sTAted aS FOllowS.