In: Advanced 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 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.