ABSTRACT
W e have discussed some fundamental properties of groups inChapter 8. Now we turn to the discussion of two types of algebraic systems that have two operations-rings and fields. The concepts of rings and fields are not entirely new to us. We have seen some concrete examples of rings and fields in the course of higher algebras, for example, the ring of integers, the field of real numbers, and the field of complex numbers. These also indicate the importance of rings and fields. In this chapter, we discuss the basic concepts of abstract rings and fields and their basic properties in general. We also analyze several important rings and fields. A ring is an algebraic system with two operations that is more complex than groups. However, in the study of some related ring theoretic problems, there are many approaches that are similar to that for groups.
