Editing Chinese remainder theorem (section)

==Generalization to arbitrary rings==
The Chinese remainder theorem can be generalized to any [[ring (mathematics)|ring]], by using [[Coprime integers#Coprimality in ring ideals|coprime ideal]]s (also called [[Ideal (ring theory)#Types of ideals|comaximal ideals]]). Two [[ideal (ring theory)|ideals]] {{mvar|I}} and {{mvar|J}} are coprime if there are elements <math>i\in I</math> and <math>j\in J</math> such that <math>i+j=1.</math> 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.<ref>{{harvnb|Ireland|Rosen|1990|page=181}}</ref><ref name=Sengupta>{{harvnb|Sengupta|2012|page=313}}</ref>

Let {{math|''I''<sub>1</sub>, ..., ''I<sub>k</sub>''}} be two-sided ideals of a ring <math>R</math> and let {{math|''I''}} be their [[intersection (set theory)|intersection]]. If the ideals are pairwise coprime, we have the [[ring isomorphism|isomorphism]]: 
:<math>\begin{align}
    R/I &\to (R/I_1) \times \cdots \times (R/I_k) \\
  x \bmod I &\mapsto (x \bmod I_1,\, \ldots,\, x \bmod I_k),
\end{align}</math>
between the [[quotient ring]] <math>R/I</math> and the [[product of rings|direct product]] of the <math>R/I_i,</math>
where "<math>x \bmod I</math>" denotes the [[image (mathematics)|image]] of the element <math>x</math> in the quotient ring defined by the ideal <math>I.</math>
Moreover, if <math>R</math> is [[commutative ring|commutative]], then the ideal intersection of pairwise coprime ideals is equal to their [[product of ideals|product]]; that is
:<math>
I= I_1\cap I_2 \cap\cdots\cap I_k= I_1I_2\cdots I_k,
</math>
if {{mvar|I{{sub|i}}}} and {{mvar|I{{sub|j}}}} are coprime for all {{math|''i'' ≠ ''j''}}.

===Interpretation in terms of idempotents===

Let <math>I_1, I_2, \dots, I_k</math> be pairwise coprime two-sided ideals with <math> \bigcap_{i = 1}^k I_i = 0,</math> and 
:<math>\varphi:R\to (R/I_1) \times \cdots \times (R/I_k)</math>
be the isomorphism defined above. Let <math>f_i=(0,\ldots,1,\ldots, 0)</math> be the element of <math>(R/I_1) \times \cdots \times (R/I_k)</math> whose components are all {{math|0}} except the {{mvar|i}}{{hairsp}}th which is {{math|1}}, and <math>e_i=\varphi^{-1}(f_i).</math>

The <math>e_i</math> are [[central idempotent]]s that are pairwise [[central idempotent|orthogonal]]; this means, in particular, that <math>e_i^2=e_i</math> and <math>e_ie_j=e_je_i=0</math> for every {{mvar|i}} and {{mvar|j}}. Moreover, one has <math display=inline>e_1+\cdots+e_n=1,</math> and <math>I_i=R(1-e_i).</math>

In summary, this generalized Chinese remainder theorem is the equivalence between giving pairwise coprime two-sided ideals with a zero intersection, and giving central and pairwise orthogonal idempotents that sum to {{math|1}}.<ref>{{harvnb|Bourbaki, N.|1989|page=110}}</ref>