ABSTRACT
18.1 We have observed the presence of a second operation, multiplication, on integral numbers. If one formalize the rules controlling the interaction between addition and multiplication then one arrives at the concept of ring. The difference with the multiplication of integers is that this operation is not derived from the addition. Ring structures appear in nature most often via composition of endomorphisms of an object already carrying an additive structure, or point-wise multiplication of functions to objects already carrying a multiplicative structure. In this way endomorphism rings (18.6.3) and rings of matrices (19.6.4) appear. Again we provide the notion of ring homomorphisms as the maps preserving both operations involved. The kernel of a ring homomorphism is a two-sided ideal and not a subring; therefore we have to study two-sided ideals in more detail. The role played by them will become more evident in the next Chapter. Two-sided ideals are both left and right ideals of the ring, but the latter one-sided objects are important in their own right. In fact they belong to the module theory over the ring (Chapter 26), but can also be discussed in intrinsically ring theoretical terms as we do in 18.27, and following, up to Theorem 18.31 giving the correspondence theorem for subrings and ideals.
