The pair <*S*, +> is said to be a *group* if:

- for every
*a*,*b*, and*c*in*S*, (*a*+*b*) +*c*=*a*+ (*b*+*c*); - there is an identity element 0 in
*S*such that for every*a*in*S*, 0 +*a*=*a*+ 0 =*a*; and - for every
*a*in*S*, there is an inverse -*a*such that*a*+ -*a*= -*a*+*a*= 0.

If we use the symbol `*` as the operator in the group, we
usually call the identity 1 and write the inverse of *a* as
*a*^{-1}.

A group <*S*, +> is said to be *commutative* if for
every *a* and *b* in *S*, *a* + *b* =
*b* + *a*. We also call such a group an *abelian group*.

Suppose *S* has *n* elements. Then we say the group
<*S*, +> is of *order n*.

Let <*S*, +> be a group of order *n*. We
abbreviate the sum of *i* *a*'s, *a* + *a* + ... +
*a*, as *ia*. (Likewise, we abbreviate the product of
*i* *a*'s, *a* `*` *a* `*`
... `*` *a*, as *a ^{i}*.) <

Let <*S*, +> and <*S'*, +> be groups such that
*S'* is a subset of *S*. Then <*S'*, +> is said
to be a *subgroup* of <*S*, +>.

The triple <*S*, +, `*`> is said to be a *ring*
if:

- <
*S*, +> is an abelian group; - for every
*a*,*b*, and*c*in*S*, (*a*`*`*b*)`*`*c*=*a*`*`(*b*`*`*c*); and - for every
*a*,*b*, and*c*in*S*,*a*`*`(*b*+*c*) =*a*`*`*b*+*a*`*`*c*and (*b*+*c*)`*`*a*=*b*`*`*a*+*c*`*`*a*.

If there is an element 1 in *S* such that for every *a* in
*S*, 1 `*` *a* = *a* `*` 1 = *a*,
then the ring <*S*, +, `*`> is said to be a *ring
with unit element*.

If for every *a* and *b* in *S*, *a* `*`
*b* = *b* `*` *a*, then the ring <*S*,
+, `*`> is said to be a *commutative ring*.

Let 0 be the additive identity (i.e., the identity of <*S*,
+>) of the ring <*S*, +, `*`>. If
<*S*-{0}, `*`> is an abelian group, then
<*S*, +, `*`> is said to be a *field*.

Copyright © Rod Howell, 2001. All rights reserved.

Rod Howell (howell@cis.ksu.edu)