 
Summary: Chapter 5
Rings
Nothing proves more clearly that the mind
seeks truth, and nothing reflects more glory
upon it, than the delight it takes, sometimes
in spite of itself, in the driest and thorniest
researches of algebra.
 Bernard de Fontenelle
This chapter introduces what is in some ways the next
logical structure in modern or abstract algebra: the
ring. The ring we know best at this point is the inte
gers under addition and multiplication. It is in many
ways the quintessential ring, in the sense that the
strangeness of other rings is measured by the proper
ties they fail to share with the integers.
5.1 Definitions and Examples
Definition 5.1 A ring is a set R with two opera
tions, called addition, denoted +, and multiplica
tion, denoted · or ×, which obey the following rules:
· (R, +) is a commutative group.
