《Discrete Mathematics II教学-华南理工》Section 15 Rings and Fields.pdfVIP

IV.18 Rings and Fields 1 Part IV. Rings and Fields Section IV.18. Rings and Fields Note. Roughly put, modern algebra deals with three types of structures: groups, rings, and fields. In this section we introduce rings and fields. In this last part of the course, we look at some properties of these algebraic structures. We will finally see the “basic goal” of the text in Section IV.22 when mention is made of solving polynomial equations in a field (see page 206). Note. In Introduction to Modern Algebra 2 (MATH 4137/5137) you will explore rings some more (Parts V and IX) and you will explore fields a lot more (in Parts VI and X). There are still some important results from group theory yet to be presented and these can be found in Part VII. Applications of group theory to topology can be found in the optional Part VIII. Note. Rings and fields have two binary operations. We denote these operations as + and ·. In group theory, we used both + and · as the binary operation of a group. The choice of + or · was irrelevant in the group setting; it was usually motivated by the types of example under consideration (Zn is additive and GL(n, R) is multiplicative), but in group theory the difference between + and · is purely notational. This is not the case in ring theory or field theory. We require, by definition, different properties for one binary operation (+) than for the other (·). IV.18 Rings and Fields 2 Definition 18.1. A ring R, +, · is a set R together with two binary operations + and ·, called addition and multiplication, respectively, defined on R such that: R1: R, + is an abelian group. R2 : Multiplication · is associative: (a · b) · c = a · (b · c) for all a, b, c