Chinese remainder theorem

Let R be a commutative ring with identityPlanetmathPlanetmathPlanetmath. If I1,,In are ideals of R such that Ii+Ij=R whenever ij, then let


The sum of quotient maps R/IR/Ii gives an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


This has the slightly weaker consequence that given a system of congruencesMathworldPlanetmathPlanetmathPlanetmathPlanetmath xai(modIi), there is a solution in R which is unique mod I, as the theorem is usually stated for the integers.

Title Chinese remainder theoremMathworldPlanetmathPlanetmathPlanetmath
Canonical name ChineseRemainderTheorem1
Date of creation 2013-03-22 12:16:43
Last modified on 2013-03-22 12:16:43
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 7
Author bwebste (988)
Entry type Theorem
Classification msc 11N99
Classification msc 11A05
Classification msc 13A15
Related topic ChineseRemainderTheoremInTermsOfDivisorTheory