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单词 ChineseRemainderTheorem1
释义

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

I=i=1nIi=i=1nIi.

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

R/Ii=1nR/Ii.

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.

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更新时间:2025/5/4 19:04:39