Chinese remainder theorem for rings, noncommutative case
Clearly is a homomorphism with kernel . It remains to show the surjectivity.
Continuing, we obtain that . We show similarly that:
Given elements , we can find and such that .
and we conclude that is surjective as required.∎
Notes 1.The relation is satisfied when is ring with unity. In that case .
2. The Chinese Remainder Theorem (http://planetmath.org/ChineseRemainderTheorem) case for integers is obtained from the above result. For this, take and . The fact that two solutions of the set of congruences must is a consequence of:
|Title||Chinese remainder theorem for rings, noncommutative case|
|Date of creation||2013-03-22 16:53:45|
|Last modified on||2013-03-22 16:53:45|
|Last modified by||polarbear (3475)|
|Synonym||chinese remainder theorem|