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Solve Problem #3
Let L, K be ideals of R with L C K then we have a well defined homomorphism PL,K: R/ L + R/K, [a][ [a]k Theorem 0.1 (Chinese Remainder Theorem for rings). Let I and J be coprime ideals of a ring R. Then a (pin),I, PINJ,): R/IN J + R/ I x R ) J is an isomorphism. 1 Example 0.2. (Does not need to be handed in) Check that in the ring Z this theorem reduces to the Chinese remainder theorem. 3. Problem Prove the Chinese Remainder Theorem for Rings