We generalize the algebraic structure of
*addition* and *multiplication* over the
integers.^{1}

A ring (or ring with identity) $(R, f, g)$ is a set $A$ and two binary operations on $R$ satisfying the following set of conditions.

(A)
(i) $f$ is *associative*.
(ii) $f$ is *commutative*,
(iii) $A$ has an *identity element* for
$f$ (i.e., there is $e \in R$ with $f(r, e) =
f(e, r) = r$ for all $r \in R$
(iv) $R$ has *inverse elements* for $f$
(i.e., for any $r \in R$, there is $\tilde{r}
$ satisfying $f(r,\tilde{r}) = f(\tilde{r}, r) =
e$)

(B)
(i) $g$ is *associative*;
(ii) $R$ has an *identity element* for $g$
(i.e., for any $r \in R$, there is $\tilde{e}
\in A$ satisfying $g(r, \tilde{e}) =
g(\tilde{e},r) = r$)

(C)
(i) $g$ *left distributes*:

\[ g(f(x, y), \alpha ) = f(g(\alpha ,x), g(\alpha ,y)) \quad \text{for all } x, y, \alpha \in R \]

(ii) $g$\[ g(\alpha , f(x,y)) = f(g(\alpha ,x), g(\alpha ,y)) \quad \text{for all } x, y, \alpha \in R \]

Conditions (A) concern $f$, conditions (B) concern $g$, and conditions (C) relate the two.

Clearly, $\Z $ with addition and multiplication is a ring. The element referred to in (A.2) is $0 \in \Z $, so we refer to this element in any ring as the additive identity. That referred to (A.3) is $1 \in \Z $, so we refer to this element in any ring as the multiplicative identity. We refer to the elements mentioned in (A.4) as additive inverses. We call to $f$ ring addition and $g$ ring multiplication.

A ring which for which multiplication is
commutative is called a
commutative ring.
Note that a ring is *always* commutative
with respect to addition, here the term
commutative refers to multiplication.
A ring for which there are inverse elements,
excepting 0, is called a
division ring).

Of course, the integers form a ring with the usual notion of addition and multiplication. For another trivial example, consider $\set{0}$ with $0+0 = 0$ and $0\cdot 0 = 0$; this is called the zero ring (any ring isomorphic to this one is called a trivial ring or zero ring).

The notation commonly adopted in discussing rings relies on analogy with the set of integers $\Z $. We denote the ring addition by $+$ and ring multiplication by $\cdot $. Moreover, we denote the ring’s additive identity by $0$ and the ring’s multiplicative identity by $1$. Finally, we denote the additive inverse of $a \in A$ by $-a$.

Rewriting the conditions (A), (B), (C) in this notation gives familiar-looking relations, from when the objects involved were integers. (A) (1) $a+(b + c) = (a+b)+c$; (2) $a+b = b+a$; (3) $a + 0 = 0 + a = a$; (4) $a + (-a) = 0$. (B) (1) $a(bc) = (ab)c$; (2) $1a = a1 = a$. (C) (1) $(a+b)c = ac + bc$; (2) $c(a+b) = ca + cb$.

We need not require that $0x = 0$, because we can deduce it:

\[ 0x + x = (0 + 1)x = 1x = x. \]

Similarly, $(-a)b = -(ab)$ since\[ ab + (-a)b = (a + (-a))b = 0b = 0. \]

Other familiar relations among the integers, e.g. $(-a)(-b) = ab$, may be deduced.- Future editions will likely modify this sheet, and give a genetic treatment involving the solution of polynomial equations by Galois. ↩︎