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" or "college algebra" which teaches the correct rules for manipulating formulas and algebraic expressions.
Historically, algebraic structures usually arise first in some other field of mathematics, are specified axiomatically, and 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:
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 morphism, form categories, and category theory frequently provides the formalism for translating between and comparing different algebraic structures.