Chinese remainder theorem for rings, noncommutative case
Theorem 1.
(Chinese Remainder Theorem) Let be a ring and pairwise comaximal (http://planetmath.org/Comaximal) ideals such that for all . The homomorphism:
is surjective and .
Proof.
Clearly is a homomorphism with kernel . It remains to show the surjectivity.
We have:
Moreover,
Continuing, we obtain that . We show similarly that:
Given elements , we can find and such that .
Take .
Hence
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 |
---|---|
Canonical name | ChineseRemainderTheoremForRingsNoncommutativeCase |
Date of creation | 2013-03-22 16:53:45 |
Last modified on | 2013-03-22 16:53:45 |
Owner | polarbear (3475) |
Last modified by | polarbear (3475) |
Numerical id | 16 |
Author | polarbear (3475) |
Entry type | Theorem |
Classification | msc 13A15 |
Classification | msc 11D79 |
Synonym | chinese remainder theorem |