Chinese Remainder Theorem - Statement For General Rings

Statement For General Rings

The general form of the Chinese remainder theorem, which implies all the statements given above, can be formulated for commutative rings and ideals. If R is a commutative ring and I₁, …, I_k are ideals of R which are pairwise coprime (meaning that for all ), then the product I of these ideals is equal to their intersection, and the quotient ring R/I is isomorphic to the product ring R/I₁ × R/I₂ × … × R/I_k via the isomorphism

such that

Here is a version of the theorem where R is not required to be commutative:

Let R be any ring with 1 (not necessarily commutative) and be pairwise coprime 2-sided ideals. Then the canonical R-module homomorphism is onto, with kernel . Hence, (as R-modules).