Introduction

Modern algebra is a study of sets with operations defined on them. It begins with the observation that certain familiar rules hold for different operations on different sets.

Let us consider the set of natural numbers .
The operations of addition and multiplication satisfy the following rules:

as well as

Also there is a distinguished element 1, which has the property

In the sets of integers , rationals , reals
and complex numbers all the above rules remain valid.
In addition, in each of them there is a distinguished element
such that

Also, for every element there is an element (its negative) such that

Moreover, in , and , for every there is an element such that

At this stage you should note that the following pairs of rules are very similar: (1) and (2); (3) and (4); (6) and (7); (9) and (10). The only difference is that they refer to different operations.

Our next example is the set of all matrices with real number entries. Again, we can add and multiply matrices, and we have the following rules:

This time, however, we do not have . Also, we have distinguished matrices (the zero matrix) and (the identity matrix) such that

For every matrix there exists a matrix such that

However, it is not true that for every matrix there exists a matrix such that ; such a matrix exists if and only if is invertible (i.e. if has a non-zero determinant).

Let us now fix a set , and consider the set of
all mappings
. For such a mapping
and , we shall write the image of under as .
We can compose two mappings and by applying first
one and then the other; the resulting mapping is denoted by
. Thus

(Note that this means that we multiply mappings, slightly unnaturally, from right to left. In some books you will find mappings written to the right of their argument, i.e. instead of ; a benefit of doing this is that the composition law becomes .)

Straight from the definition it follows that

and that there exists a mapping (the identity mapping, sending every to itself) satisfying

However, does not hold in general, and it is not true that for every mapping there exists a mapping such that .

In algebra, one studies an abstract set with one or more operations defined on it. These operations are assumed to satisfy some basic properties, and the aim is to study the consequences of these properties.

Edmund F Robertson

11 September 2006