ABSTRACT
20.1 Given two rings, we can define on the Cartesian product of these sets first the structure of an abelian (additive) group and then also a ring structure, in such a way that it satisfies a universal property with respect to ring homomorphisms. The idea of direct product of rings may be combined with the construction of quotient rings i.e. what is the relation between the quotient ring with respect to an intersection of two-sided ideals and the product of the quotient ring for each two-sided ideal separately? In case the two-sided ideals are relatively prime, that is their sum is the whole ring, the behaviour of the quotient structures is most satisfactory and it is fully expresses in the “Chinese Remainder Theorem” (Theorem 20.25 and Corollary 20.26). We include further applications to the Euclidian division algorithm in ℤ (Theorem 20.30). Finally we include a general version of the Chinese Remainder Theorem in case finitely many two-sided ideals, i.e. more than two, are being considered (Theorem 20.34).
