- 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 from $F^2$ to $F$ called
*addition*and*multiplication*; as usual we shall denote the images of $(a,b)$ under these two functions by $a+b$ and $a \times b$ (or $a \cdot b$, 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 that $(F,0,-,+)$ is aiv) For all $a,b$ in $F\!$, we have $a+b = b+a$ [i.e., addition is commutative]group. Familiar consequences are the right and leftcancellation rules: if, for any $a,b,c$ in $F\!$, we have $a+c=b+c$ or $c+a=b+a$, then $a=b$. This is proved by adding $(-c)$ to both sides from the right or left respectively. In particular, $a+a=a$ if and only if $a=0$. Likewise, for any $a,b$ in $F\!$, the equation $a+x=b$ has the unique solution $a = b+(-a)$, usually abbreviated $x = b - a$ (do not confuse this binary operation of “subtraction” with the unary additive inverse!). Another standard consequence of (iii) is that, for any $a_1,a_2,\ldots,a_n$ in $F\!$, the sum $a_1 + a_2 + \cdots + a_n$ is the same no matter how it is parenthesized. (In how many wayscanthat expression be parenthesized?)

Conditions (i), (ii), (iii), (iv) assert that $(F,0,-,+)$ is av) For all $a$ in $F\!$, we have $a \times 1 = 1 \times a = a$ [i.e., $1$ is a multiplicative identity]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 $a$ in $F\!$, we have $a \times a^{-1} = a^{-1} \times a = 1$ [this is what “multiplicative inverse” means]

vii) For all $a,b,c$ in $F\!$, we have $a(bc)=(ab)c$ [i.e., multiplication is associative]

In particular, restricting (v), (vi) and (vii) to $F^*\!$, we are asserting that $(F^*, 1, {}^{-1}, \times)$ is a group. You probably know already that “$a^{-1}$” is a special case of the notation $a^n$ for any integer $n$, with $a^1 = a$ and $a^{m+n} = a^m a^n$ for all integers $m,n$ (which implies $a^0 = 1$ in at least two ways); that is, in any group $G$, if we fix a group element $a$ then the map $m \mapsto a^m$ gives aviii) For all $a,b$ in $F\!$, we have $ab=ba$ [i.e., multiplication is commutative]group homomorphismfrom $({\bf Z}, +)$ to $G$.

So, the group $(F^*, 1, {}^{-1}, \times)$ is also abelian. For $a,b \in F$ with $a \neq 0$, the unique solution $a^{-1}\!b$ of $ax=b$ is usually written $b/a$ix) For all $a,b,c$ in $F\!$, we have $a(b+c)=(ab)+(ac)$ and $(a+b)c=(ac)+(bc)$. [distributive law. The second part is of course redundant by commutativity; it is required in the “skew” case (see below).](or $\frac b a$); in particular $a^{-1}$ is also $1/a$. Do not use the notation $b/a$ in a non-commutative group, because of the ambiguity between $a^{-1}\!b$ and $ba^{-1}$.

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

For all $a$ in $F\!$, we have $a \times 0 = 0 \times a = 0$.

Thus also:

For all $a,b$ in $F\!$, we have $ab=0$ if and only if $a=0$ or $b=0$ (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* $0\neq1$,
we obtain the structure of a commutative *ring* with unity.
For example, $\bf Z$ 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 $a+bi+cj+dk$
whose coefficients $a,b,c,d$ 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 **not** $F-\{0\}$ but the set of
(multiplicatively) invertible elements of $F\!$.
So if $F$ *is* a field the two notations coincide.
Whether or not $F$ is a field, $F^*$ is called the
*multiplicative group* of $F\!$.
(One sometimes also sees the equivalent term “unit group”,
because the invertible elements of a ring are called its “units”.)
You should verify that it is indeed a group. For example,
the multiplicative group in the ring of Hamilton quaternions with
integer coefficients consists of the 8 elements $\pm 1,\pm i, \pm j, \pm k$;
it is known as the “quaternion group”.
How large is the unit group of the Hurwitz quaternions?

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