Chinese Remainder Theorem - Statement For Principal Ideal Domains

Statement For Principal Ideal Domains

For a principal ideal domain R the Chinese remainder theorem takes the following form: If u₁, …, u_k are elements of R which are pairwise coprime, and u denotes the product u₁…u_k, then the quotient ring R/uR and the product ring R/u₁R× … × R/u_kR are isomorphic via the isomorphism

such that

This map is well-defined and an isomorphism of rings; the inverse isomorphism can be constructed as follows. For each i, the elements u_i and u/u_i are coprime, and therefore there exist elements r and s in R with

Set e_i = s u/u_i. Then the inverse of f is the map

such that

$g(a_1 + u_1R, \ldots, a_k + u_kR) = \left( \sum_{i=1}^k a_i \frac{u}{u_i} \left_{u_i} \right) + uR \quad\mbox{ for all }a_1, \ldots, a_k \in R.$

This statement is a straightforward generalization of the above theorem about integer congruences: the ring Z of integers is a principal ideal domain, the surjectivity of the map f shows that every system of congruences of the form

can be solved for x, and the injectivity of the map f shows that all the solutions x are congruent modulo u.

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