Rings

Abstract

Another fundamental concept in the study of algebra is that of a ring. The problem of classifying all rings (in a given class) up to isomorphism is far more complicated than the corresponding problem for groups. It will be partially dealt with in Chapter IX. The present chapter is concerned, for the most part, with presenting those facts in the theory of rings that are most frequently used in several areas of algebra. The first two sections deal with rings, homomorphisms and ideals. Much (but not all) of this material is simply a straightforward generalization to rings of concepts which have proven useful in group theory. Sections 3 and 4 are concerned with commutative rings that resemble the ring of integers in various ways. Divisibility, factorization, Euclidean rings, principal ideal domains, and unique factorization are studied in Section 3. In Section 4 the familiar construction of the field of rational numbers from the ring of integers is generalized and rings of quotients of an arbitrary commutative ring are considered in some detail. In the last two sections the ring of polynomials in n indeterminates over a ring R is studied. In particular, the concepts of Section 3 are studied in the context of polynomial rings (Section 6).