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

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Advance Math

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 thatany two solutions of the system are congruent modulo I ?J. (c) LetI and J be ideals in a ring R such that I + J = R. Show that thereexists a ring isomorphism R/(I ?J) ? = R/I ×R/J.

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