62 I The Expanding Universe of Numbers
manner just described. It was proved by Stone (1936) that every Boolean ring may
be obtained in this way. Thus the algebraic laws of set theory may be replaced by the
more familiar laws of algebra and all such laws are consequences of a small number
among them.
We now return to arbitrary rings. In the same way as forZ,inanyringRwe have
a 0 = 0 = 0 a for everyaand
(−a)b=−(ab)=a(−b) for alla,b.It follows thatRcontains only one element if 1=0. We will say that the ringRis
‘trivial’ in this case.
SupposeRis a nontrivial ring. Then, viewingRas a group under addition, the
cyclic subgroup〈 1 〉is either infinite, and isomorphic toZ/ 0 Z, or finite of orders,and
isomorphic toZ/sZfor some positive integers. The ringRis said to havecharacter-
istic0 in the first case andcharacteristic sin the second case.
For any positive integern, write
na:=a+···+a (nsummands).IfRhas characteristics>0, thensa=0foreverya∈R,since
sa=( 1 +···+ 1 )a= 0 a= 0.On the other hand,n 1 =0 for every positive integern<s, by the definition of
characteristic.
An elementaof a nontrivial ringRis said to be ‘invertible’ or aunitif there exists
an elementa−^1 such that
a−^1 a= 1 =aa−^1.The elementa−^1 is then uniquely determined and is called theinverseofa.For
example, 1 is a unit and is its own inverse. Ifais a unit, thena−^1 is also a unit and
its inverse isa.Ifaandbare units, thenabis also a unit and its inverse isb−^1 a−^1 .It
follows that the setR×of all units is a group under multiplication.
A nontrivial ringRin which every nonzero element is invertible is said to be a
division ring. Thus all nonzero elements of a division ring form a group under multipli-
cation, themultiplicative groupof the division ring. Afieldis a commutative division
ring.
A nontrivial commutative ringRis said to be anintegral domainif it has no
‘divisors of zero’, i.e. ifa=0andb=0implyab=0. A division ring also has
no divisors of zero, since ifa=0andb=0, thena−^1 ab=b=0, and henceab=0.
As examples, the setQof rational numbers, the setRof real numbers and the set
Cof complex numbers are all fields, with the usual definitions of addition and mul-
tiplication. The setHof quaternions is a division ring, and the setZof integers is an
integral domain, but neither is a field.