Algebraic Structures

To this point in the blog, most of the work has been concentrated on group theory, but really there are a lot of algebraic structures. What differentiates algebraic structures are the axioms (or rules) that we used to define them. The definition of a group provided four axioms - closure, associativity, identity, and inverses - and every other algebraic structure is defined similarly. Really, you can start with an arbitrary set and define any axioms that you want on the set and work with it and see where you get, and abstract algebra does just that but with the most useful and lucrative sets of axioms.

In my opinion, there are three important elementary algebraic structures, which are groups, rings, and fields. I'm going to define them and discuss some examples, but I'm not going to work through them as rigorously as we've covered groups - at least not yet.

Groups

We've already discussed groups in depth, but for reference I'm going to include a more formal definition below.

Definition: Group
Let $G$ be any set and let $\cdot$ be a binary operation defined on the elements of $G$. Then $G$ is a group if it satisfies the following four axioms:

Closure

$a \cdot b \in G \; \forall a,b \in G$

Associativity

$\left ( a \cdot b \right ) \cdot c = a \cdot \left ( b \cdot c \right ) \; \forall a,b,c \in G$

Identity

$\exists e \in G \; \text{such that} \; a \cdot e = a = e \cdot a \; \forall a \in G$

Inverses

$\forall a \in G \; \exists a^{-1} \in G \; \text{such that} \; a \cdot a^{-1} = e = a^{-1} \cdot a$

Rings

Rings are a step up from groups in that rings have two binary operations - addition and multiplication. The definition is given below.

Definition: Ring
Let $R$ be any set with two binary operations on the elements of $R$, called addition (denoted by $+$) and multiplication (denoted by $\cdot$). Then $R$ is a ring if it satisfies the following nine axioms.

Additive Closure

$a + b \in R \; \forall a,b \in R$

Additive Associativity

$\left ( a + b \right ) + c = a + \left ( b + c \right ) \; \forall a,b,c \in R$

Additive Identity

$\exists 0 \in R \; \text{such that} \; a + 0 = a = 0 + a \; \forall a \in R$

Additive Inverses

$\forall a \in R \; \exists -a \in R \; \text{such that} \; a + \left ( -a \right ) = 0 = \left ( -a \right ) + a$

Additive Commutivity

$a + b = b + a \; \forall a,b \in R$

Multiplicative Closure

$a \cdot b \in R \; \forall a,b \in R$

Multiplicative Associativity

$\left ( a \cdot b \right ) \cdot c = a \cdot \left ( b \cdot c \right ) \; \forall a,b,c \in R$

Left Distribution

$a \cdot \left ( b + c \right ) = \left ( a \cdot b \right ) + \left ( a \cdot c \right ) \; \forall a,b,c \in R$

Right Distribution

$\left ( a + b \right ) \cdot c = \left ( a \cdot c \right ) + \left ( b \cdot c \right ) \; \forall a,b,c \in R$

I know that it looks like a big nasty definition and nine axioms seem like a lot, but its not so bad when you break it down. The five additive axioms tell us that $R$ is an abelian group under addition. The multiplicative axioms give us what we're allowed to do with the multiplication operator. Notice that $R$ does not include either a multiplicative identity or multiplicative inverses. The two distributive laws give the ways that addition and multiplication interact. A ring is called commutative if elements of $R$ commute under multiplication.

Fields

Fields are very important in many areas of mathematics, including some advanced group theory, linear algebra, and some areas of topology. Fields - especially algebraically closed fields like $\mathbf{C}$ - give basically all of the properties that we need to support other, more complicated systems like vector spaces, modules, or polynomials. The definition is given below.

Definition: Field
Let $F$ be a ring. Then $F$ is a field if, in addition to the ring axioms, it also satisfies the following axioms.

Multiplicative Commutivity

$a \cdot b = b \cdot a \; \forall a,b \in F$

Unity (Multiplicative Identity)

$\exists 1 \in F \; \text{such that} \; 0 \neq 1 \; \text{and} \; a \cdot 1 = a = 1 \cdot a \; \forall a \in F$

Multiplicative Inverses

$\forall a \in F \; \text{with} \; a \neq 0, \; \exists a^{-1} \in F \; \text{such that} \; a \cdot a^{-1} = 1 = a^{-1} \cdot a$

A field, $F$, is automatically an abelian additive group since it inherits the ring axioms (from the first line of the definition) and the field axioms give a multiplicative identity, multiplicative commutivity, and multiplicative inverses (except that $0$ has no inverse). This gives the abelian multiplicative group structure of $F\setminus \left \{ 0 \right \}$ (the field without the additive identity). And, of course, $F$ still has the distributive properties. Now that we're in a field we get all of the addition, subtraction, multiplication, and division properties as well as commutivity and distribution. This means that with respect to the usual operations of addition and multiplication we finally get to do all the things that we think we should be able to do, which is nice in vector spaces, topological spaces, and modules.

In the last entry we looked at a few number systems, and now we know some of the basic algebraic structures, and next time we'll see how they fit together - or rather how they fail to fit together.

References

Previous Related Post: Numbers

Wikipedia: Group

Wikipedia: Ring

Wikipedia: Field