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:
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
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:
Let us now fix a set , and consider the set of
. 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
Straight from the definition it follows 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