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Generalization to arbitrary rings

The Chinese remainder theorem can be generalized to any ring, by using coprime ideals (also called comaximal ideals). Two ideals I and J are coprime if there are elements i ∈ I {\displaystyle i\in I} and j ∈ J {\displaystyle j\in J} such that i + j = 1. {\displaystyle i+j=1.} 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.

Let I,..., I be two-sided ideals of a ring R {\displaystyle R} and let I be their intersection. If the ideals are pairwise coprime, we have the isomorphism:

between the quotient ring R / I {\displaystyle R/I} and the direct product of the R / I i, {\displaystyle R/I_{i},} where " x mod I {\displaystyle x{\bmod {I}}} " denotes the image of the element x {\displaystyle x} in the quotient ring defined by the ideal I. {\displaystyle I.} Moreover, if R {\displaystyle R} is commutative, then the ideal intersection of pairwise coprime ideals is equal to their product; that is

if I and I are coprime for all i ≠ j.