Abstract Algebra II
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 8 (12,13,15,16,17,18) (General algebraic systems)

Prerequisites:

# Getting Oriented

 Rough Guides to Algebra
 General Algebra Special Topics

There are many algebraic structures which may be given to sets in order to solve problems and bring out some extraordinary properties. Among these are groups, rings, fields, vector fields, modules, algebras, categories, and of course all the finer structures (subrings, domains, and the like). As one might guess, abstract algebra is very heavy in definitions, but each of these structures has its own place, and each is widely applicable in many fields of mathematics.

We will begin with a survey of the various structures, so that one may see their similarities and differences. This will include the definition of a category, which allows for some definitions not specific to a single structure. We will then focus on each of the structures and their importance.

# Various Structures

## The Basic Structure

We'll begin with the structure that takes its name from the subject: an algebra, or a set with an operation law:

Algebra
a

In undergraduate abstract algebra, one encounters primarily just three of the structures: groups, rings, and fields. In a nutshell, a group is a set with one operation (say addition/subtraction), a ring is one with two (addition/subtraction and multiplication), and a field is a ring which additionally allows division. Examples of these include $\mathbb{Z}$ for the first two cases, and $\mathbb{R}$ or $\mathbb{C}$ for the third case. Let's give the formal definitions:

Group
a set $G$ with operation $+$ which is associative ($a+(b+c)=(a+b)+c$), has an identity ($\exists e\in G$ s.t. $a+e=e+a=a$), and provides inverses $-a$ (for all $a\in G$ we have $a+(-a)=(-a)+a=e$).

Important types of groups include abelian (addition is commutative), cyclic (generated by a single element)

Ring
a group $R$ with additional associative operation * with the distributive property ($(a+b)c=ac+bc$ and $a(b+c)=ab+ac$).

Rings may be specified to be commutative or to have a unity.

Types of rings include integral domains (there are no zero-divisors and the cancellation law holds), unique factorization domains (UFD's), principal ideal domains (PID's, when every ideal is generated by a single element), and Euclidean domains (ED's, which have a division algorithm). These can also be viewed as intermediate steps between rings and fields.

Field
a ring $F$ for which the set of nonzero elements $F^*$ also forms a commutative group under multiplication.

Fields have all the nice properties mentioned above; an algebraically closed field is even nicer, contains the roots of all its polynomials.

A last important structure to point out is the ring of polynomials. Given any commutative ring $R$, the set of (finite) polynomials with coefficients in $R$ is also a commutative ring, denoted $R[x]$. Very often, properties of $R$ carry over to $R[x]$: for example, if $R$ is an integral domain, then $R[x]$ is as well. Polynomial rings are infinitely important in ring and field theory, and prove quite useful in classifying these structures.