a. State and prove the generalization of Example 18.15 for a direct product with n factors. 

b. Prove the Chinese Remainder Theorem: Let a_i, b_i ∈ Z+ for i = 1, 2, ···, n and let gcd(b_i, b_j) = 1 for i ≠ j. Then there exists x ∈ Z⁺ such that x = a_i (mod b_i) for i = 1, 2, ···· , n.

Data from in Example 18.15 

We claim that for integers r ands where gcd(r, s) = 1, the rings Z_rs and Z_r x Z_s are isomorphic. Additively, they are both cyclic abelian groups of order rs with generators 1 and (1, 1) respectively. Thus ∅ : Z_rs → Z_r x Z_s defined by ∅(n · 1) = n · (1, 1) is an additive group isomorphism. To check the multiplicative Condition 2 of Definition 18.9, we use the observation preceding this example for the unity (1, 1) in the ring Z_r x Z_s, and compute. 

(∅(nm)= (nm)· (1, 1) = [n · (1, 1)][m · (1, 1)] = ∅(n)∅(m).