PlanetMath: Chinese remainder theorem

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Chinese remainder theorem

(Theorem)

Let be a commutative ring with identity. If $I_1,\ldots,I_n$ are ideals of such that whenever $i\neq j$ , then let

$\displaystyle I=\cap_{i=1}^n I_i = \prod_{i=1}^n I_i.$

The sum of quotient maps $R/I\to R/I_i$ gives an isomorphism

$\displaystyle R/I\cong \prod_{i=1}^n {R}/{I_i}.$

This has the slightly weaker consequence that given a system of congruences $x\cong a_i\pmod{I_i}$ , there is a solution in

which is unique mod

, as the theorem is usually stated for the integers.

"Chinese remainder theorem" is owned by bwebste. [ full author list (2) | owner history (1) ]

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Cross-references: integers, solution, congruences, consequence, isomorphism, quotient maps, sum, ideals, identity, commutative ring
There are 7 references to this entry.

This is version 4 of Chinese remainder theorem, born on 2002-02-03, modified 2003-09-06.
Object id is 1729, canonical name is ChineseRemainderTheorem2.
Accessed 5253 times total.

Classification:

AMS MSC:	11N99 (Number theory :: Multiplicative number theory :: Miscellaneous)
	11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)
	13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory)

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