The Chinese Remainder Theorem for Rings. Let R be a ring and I and J be...

The Chinese Remainder Theorem for Rings.

Let R be a ring and I and J be ideals in R such that I + J = R. (a) Show that for any r and s in R, the system of equations x ≡ r (mod I) x ≡ s (mod J) has a solution. (b) In addition, prove that any two solutions of the system are congruent modulo I ∩J. (c) Let I and J be ideals in a ring R such that I + J = R. Show that there exists a ring isomorphism R/(I ∩J) ∼ = R/I ×R/J.

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Colby Messinger answered 1 year ago

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