Abstract algebra

Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from "elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers.

Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics.

Examples of algebraic structures with a single binary operation are:

• semigroups
• monoids
• groups
• quasigroups

More complicated examples include:

• rings and fields
• modules and vector spaces
• associative algebras and Lie algebras
• lattices and Boolean algebras

In universal algebra, all those definitions and facts are collected that apply to all algebraic structures alike. All the above classes of objects, together with the proper notion of homomorphism, form categories, and category theory frequently provides the formalism for translating between and comparing different algebraic structures.

External references:

• John Beachy: Abstract Algebra Online, http://www.math.niu.edu/~beachy/aaol/contents.html Comprehensive list of definitions and theorems.
• Joseph Mileti: Mathematics Museum (Abstract Algebra), http://www.math.uiuc.edu/~mileti/Museum/algebra.html A good introduction to the subject in real-life terms.