- Two distinguished elements called 0 and 1, which must be different
- A function from
*F*to*F*called*additive inverse*and denoted by the unary minus sign (so the additive inverse of*a*is −*a*), and a function from*F*−{0} to*F*−{0} called*multiplicative inverse*that takes*a*to an element called*a*^{−1}(we shall usually write*F*for^{*}*F*−{0}) - Two functions
*F*^{2}to*F*called*addition*and*multiplication*; as usual we shall denote the images of (*a,b*) under these two functions by and*a + b* (or*a*b**a·b*, or simply*ab*).

i) For all *a* in *F*, we have *a*+0=0+*a=a*
[i.e., 0 is an additive identity]

ii) For all *a* in *F*, we have
*a*+(−*a*)=(−*a*)+*a*=0
[this is what “additive inverse” means]

iii) For all *a,b,c* in *F*, we have *a+(b+c)=(a+b)+c*
[i.e., addition is associative]

Conditions (i), (ii), (iii) assert thativ) For all( is aF, 0, −, +)group. Familiar consequences are the right and leftcancellation rules: if, for anya,b,cinF, we havea+c=b+corc+a=b+a, thena=b. This is proved by adding (−c) to both sides from the right or left respectively. In particular,a+a=aif and only ifa=0. Likewise, for anya,binF, the equationa+x=bhas the unique solutionx=b+(−a), usually abbreviatedx=b−a(do not confuse this binary operation of “subtraction” with the unary additive inverse!). Another standard consequence of (iii) is that, for anyin a_{1},a_{2}, …,a_{n}F, the sumis the same no matter how it is parenthesized. (In how many ways a_{1}+a_{2}+ … +a_{n}canthat expression be parenthesized?)

Conditions (i), (ii), (iii), (iv) assert thatv) For all( is aF, 0, −, +)commutative group, a.k.a.abelian grouporadditive group. The first alias is a tribute to N. H. Abel (1802–1829); the second reflects the fact that in general one only uses “+” for a group law when the group is commutative — else multiplicative notation is almost always used.

vi) For all

vii) For all

In particular, restricting (v), (vi) and (vii) toviii) For allF, we are asserting that^{*}( is a group.F, 1,^{*}^{−1}, *)

So, the groupix) For all( is also abelian.F, 1,^{*}^{−1}, *)

From (ix) together with the additive properties follows the basic identity:

For allainF, we havea*0=0*a=0.

Thus also:

For alla,binF, we haveab=0 if and only ifa=0 orb=0 (or both).

That is, a field has no (nontrivial) zero divisors.

If *F* satisfies all the field axioms
except (viii), it is called a *skew field*;
the most famous example is the *quaternions* of
W. R. Hamilton (1805–1865). Much of linear algebra can
still be done over skew fields, but we shall not pursue this
much, if at all, in Math 55.

Note that (vi) is the only axiom using the multiplicative inverse.
If we drop the existence of multiplicative inverses and axiom (vi),
*as well as the condition* 0≠1,
we obtain the structure of a commutative *ring* with unity.
For example, **Z** is such a ring which is not a field.
A ring may have nontrivial zero divisors (we shall see an example
of this in class); if it does not, it is called a *domain*.

If we also drop axiom (viii) from the ring axioms, we have a
ring with unity which need not be commutative. An example is
the set of the Hamilton quaternions
*a*+*bi*+*cj*+*dk**a,b,c,d* are all integers.
Curiously if we allow the coefficients to be either all integers
or all

If *F* is a ring that need not be a field, the notation
*F ^{*}* means

A

- 1*
*v=v*for all*v*in*V*[so 1 remains a multiplicative identity], - for all scalars
*a,b*and all vectors*u,v*we have*a(u+v)=au+av*and*(a+b)u=au+bu*[distributive properties], - for all scalars
*a,b*and all vectors*v*we have*a(bv)=(ab)v*[associative property].

Notice that these axioms do not use the multiplicative inverse;
they can thus be used equally when *F* is any ring
(even a non-commutative ring),
in which case the resulting structure is called a *module*
over *F*. But multiplicative inverses are used to prove most of
the basic theorems on vector spaces, so those theorems do not hold
in the more general setting of modules; for instance one cannot speak
of the dimension of a general module, even over **Z**.