Abstract Algebra

Introduction

Abstract algebra is a major area within mathematics, which studies algebraic structures such as groups, rings, fields, modules, vector spaces, and algebras. It is a foundational discipline that has wide applications in many areas of mathematics and science.

A collection of mathematical symbols and equations representing abstract algebra concepts.

History

The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. The development of abstract algebra was influenced by a number of mathematical discoveries and the need for more rigorous mathematical analysis.

Basic Concepts

Groups

A group is a set of elements combined with an operation that combines any two of its elements to form a third element in such a way to satisfy four conditions called group axioms.

Rings

A ring is a set equipped with two binary operations satisfying properties similar to those of arithmetic operations.

Fields

A field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.

Modules

A module over a ring is a generalization of the notion of vector space over a field.

Vector Spaces

A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars.

Algebras

An algebra over a field is a vector space equipped with a bilinear product.

Applications

Abstract algebra has many applications in different fields of science and technology. It is used in physics, computer science, and engineering, among others.