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Miklós Schweitzer 1954- Problem 8

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September 29, 2015
Ring Theorycollege contests

Problem Statement

8. Prove the following generalization of the well-known Chinese remainder theorem: Let RR be a ring with unit element and let A1,A2,… .An(n⩾2)A_{1},A_{2},\dots . A_{n} (n\geqslant 2) be pairwise relative prime ideals of RR. Then, for arbitrary elements c1,c2,…,cnc_{1},c_{2}, \dots , c_{n} of RR, there exists an element x∈Rx\in R such that x−ck∈Ak(k=1,2,…,n)x-c_{k} \in A_{k} (k= 1,2, \dots , n). (A. 17)
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