Why Read This Book? - Index of

Rings
9
We can create algebraic structures of greater complexity than a group by endowing
a set with two binary operations and laying down some assumptions about how
these operations behave, both on their own and in relation to each other. In this
chapter we look at several such structures. Before we do, some explanation is in
order about how we will proceed, for the theory of rings involves so many details
that a road map will be very helpful.
First, in Section 9.1, we will define the most general algebraic structure with
two binary operations, a ring, and construct several important examples. At the
same time, we will make a passing reference to fields, the most specialized kind
of ring we will study. In Section 9.2, we define subring and construct a number of
examples. In Section 9.3, we will look at several properties that the most general
rings share. One particularly important class of rings can be created by adjoining
an element to a given ring; we devote Section 9.4 to this class of examples.
In Section 9.5, we dive down inside a ring to look at specialized substructures
of a general ring. Ideals, principal ideals, prime ideals, and maximal ideals are
special types of substructures we will see there. In Sections 9.8–9.11, we will
study four increasingly specialized kinds of rings: integral domains, unique factorization
domains, principal ideal domains, and Euclidean domains. Each class
of these structures is a proper subset of the class that comes before it, so as
we progress, we will demonstrate (or at least refer to) examples that illustrate
this. For example, we will see a ring that is not an integral domain, an integral
domain that is not a unique factorization domain, and so on. In Section 9.14 we
will look at ring morphisms, and finally, in Section 9.15, we will build quotient
rings.
9.1 Rings and Fields
9.1.1 Rings Defined
The simplest structure with two binary operations—and therefore the point
where we begin—is called a ring. Because the assumptions we make about ring
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