Chinese remainder theorem in terms of divisor theory
In a ring with a divisor theory, a congruence with respect to a divisor
module (http://planetmath.org/Congruences) that .
Theorem. Let be an integral domain having the divisor theory . For arbitrary pairwise coprime divisors in and for arbitrary elements of the domain there exists an element in such that
Proof. Let
Apparently, the divisors are mutually coprime, whence there are in the ring the elements divisible by the divisors , respectively, such that
(1) |
For every , the divisor divides and therefore also the element . Then the equation (1) implies that and thus the element
satisfies
for each . Q.E.D.
References
- 1 М. М. Постников:Введение в теорию алгебраических чисел. Издательство ‘‘Наука’’. Москва (1982).