Chinese remainder theorem in terms of divisor theory

In a ring with a divisor theory, a congruenceMathworldPlanetmathαβ(mod𝔞)  with respect to a divisorMathworldPlanetmathPlanetmath module ( 𝔞 that  𝔞α-β.

Theorem.  Let 𝒪 be an integral domainMathworldPlanetmath having the divisor theory  𝒪*𝔇.  For arbitrary pairwise coprime divisors 𝔞1,,𝔞s  in 𝔇 and for arbitrary elements  α1,,αs  of the domain 𝒪 there exists an element ξ in 𝒪 such that

{ξα1(mod𝔞1)    ξαs(mod𝔞s)

Proof.  Let


Apparently, the divisors  𝔟1,,𝔟s  are mutually coprime, whence there are in the ring 𝒪 the elements  β1,,βsdivisible by  the divisors  𝔟1,,𝔟s,  respectively, such that

β1++βs=1. (1)

For every  ij,  the divisor 𝔞i divides 𝔟j and therefore also the element βj.  Then the equation (1) implies that  βi1(mod𝔞i) and thus the element




for each  i=1,,s.  Q.E.D.


  • 1 М. М. Постников: Введение  в  теорию  алгебраических  чисел.  Издательство  ‘‘Наука’’. Москва (1982).
Title Chinese remainder theorem in terms of divisor theory
Canonical name ChineseRemainderTheoremInTermsOfDivisorTheory
Date of creation 2013-03-22 18:01:58
Last modified on 2013-03-22 18:01:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 11A51
Classification msc 13A05
Related topic ChineseRemainderTheorem
Related topic ChineseRemainderTheorem2
Related topic CongruenceInAlgebraicNumberField
Related topic WeakApproximationTheorem