1.7. Rings and Fields 37real, complex-which we wish to consider, it is convenient to define abstract
structures which embody these.
In thinking about these abstract structures, it is usually adequate to
imagine you are dealing with some concrete model. Thus, a field is a struc-
ture which embraces as particular cases the sets of rationals, reals or com-
plex numbers, so you can think of a field as being very like any one of
these sets in the way in which the elements can be combined by addition,
subtraction, multiplication and division. However, any result which can be
established for fields in general holds for rational, real or complex numbers
in particular. Rings and integral domains, in which division is not always
possible, are exemplified by the set of integers or the set of polynomials.
Here are the axioms, or ground rules by which we shall operate.
Let 5’ be a system of entities for which there are two operations, + (which
we will call addition) and. (which we will call multiplication). Consider the
following axioms:
A.l. If a and b belong to S, then a + b belongs to S.
A.2. For a and b in S, a + b = b + a.
A.3. For a, b, c in S, (a + b) + c = a + (b + c).
A.4. There is an element in S, denoted by 0 and called the zero for which
a + 0 = 0 + a = a whenever a belongs to S.
A.5. Given any element a in S, there is exactly one element, denoted by
-a and called the additive inverse, such that a + (-u) = (-u) + a = 0.
M.l. If a, b belong to S, then ab belongs to S.M.2. For a, b in S, ub = bu.
M.3. For a, b, c in S, (ub)c = u(bc).M.4. There is an element in S, denoted by 1 and called the identity, for
which a. 1 = 1. a = a whenever a belongs to S.
M.5. For any a in S with a # 0, there is an element, denoted by a-l and
called the multiplicative inverse, such that
a.a --l=.-l.a=1D. For a, b, c in S, u(b + c) = ub + UC and (b + c)a = bu + cu.
Any system of entities which satisfies all of these axioms is called a field.
Some structures do not quite manage to be fields, such as:
ring: a system satisfying A.l-5, M.l, M.3, D.
commutative ring: a ring which satisfies M.2.
commutative ring with an identity: a ring which satisfies M.2 and M.4.
integral domain: a commutative ring with identity which has no zero
divisors (this means that if ub = 0, then either a = 0 or b = 0).