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.
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