Mathematics Department Stanford University
Math 61CM/DM – Basic algebraic structures
The purpose of this handout is to provide some basic definitions and a brief discussion of some
key ideas and objects in algebra. One main new idea you will be studying in class is the notion
of a field. This is a concept which generalizes the algebraic properties of the real numbers
(as well as the rational numbers and the complex numbers). However, mathematicians find it
useful to abstract the properties of these familiar fields into a general definition. It turns out
that there are a lot of other interesting examples; we present a few of these below. One of
the points of this is that a substantial portion of linear algebra can be done in the setting of
general fields, rather than how it is more customarily done, over the real numbers. This will be
especially pertinent to the students in 61DM. For the 61CM students, you should view this as
a good way to get accustomed to a slightly different level of abstraction, and you should also
use this as an opportunity to get used to the fact that the basic results and methods in linear
algebra are really “algebraic”, i.e., only depend on the handful of axioms in the definition of a
field.
In mathematics, the most fundamental objects are sets. These are just collections of objects.
We will be concerned with either finite or certain infinite sets. (It turns out that there is quite
a lot that can be said about sets in general, but that’s another story.) However, sets are kind
of static objects – they just sit there. One can define progressively more sophisticated objects
by prescribing certain sets of rules, or axioms, about how the elements of the sets are related
to or interact with one another. We are going to describe a few different types of rules, each
slightly more specific than the last, leading to different types of algebraic objects: semigroups,
groups and then fields. This handout collects these definitions, discusses a few examples and
basic results.
Definition 1 A semigroup with unit (G, ∗, e)is a set of objects Gcontaining a distinguished
element e∈G, along with a map ∗:G×G→G, with the following properties:
1. (Associativity) For all x, y , z ∈G,x∗(y∗z)=(x∗y)∗z.
2. (Units) There exists an element e∈Gsuch that for all x∈G,x∗e=x=e∗x.
You should think of the operation ∗as being ‘multiplication’; it is a way of combining elements
of the set Gto get other elements. Here is a basic example: let G={1,2,3, . . .}, and ∗ordinary
multiplication. The ‘identity element’ eis simply 1. It is obvious that the two rules hold. Here
is another example: now let G={0,1,2, . . .}, but this time, suppose that ∗is the symbol which
corresponds to addition. Now e= 0 and once again the two rules hold.
This is one of the most primitive algebraic structures: we have a set and a way of combining
elements in it. We only have one “extra”, which is the existence of a unit e. (One can also talk
about semigroups without units; there are even some interesting examples!)
Lemma 1 In any semigroup with (G, ∗, e), the unit eis unique.
Proof: Suppose e, f ∈Gare two possibly different elements which both satisfy the properties
of a unit. Then e=e∗fsince fis a unit, and e∗f=fsince eis a unit. Combining these,
we see that e=f.This means that one can talk about the unit, i.e. given (1) and (2), e
is unique.
It is a bit more interesting if one adds one further property:
Definition 2 A group (G, ∗)is a semigroup with unit (G, ∗, e)with one additional property:
