Algebraic structures are fundamental objects in mathematics, providing a framework for studying various mathematical concepts. They consist of a set of elements together with one or more operations defined on the set. This page provides an overview of some common algebraic structures.
Types of Algebraic Structures
Groups
- A group is a set equipped with a single binary operation that combines any two elements to form a third element.
- The operation must satisfy four axioms: closure, associativity, identity, and inverse.
- Learn more about groups.
Rings
- A ring is an algebraic structure consisting of a set equipped with two binary operations, addition and multiplication.
- The operations must satisfy certain axioms, including closure, associativity, commutativity, and distributivity.
- Explore rings in more detail.
Fields
- A field is a ring in which every nonzero element has a multiplicative inverse.
- Fields are important in various areas of mathematics, including number theory and algebra.
- Discover more about fields.
Modules
- A module is an algebraic structure that extends the concept of a vector space.
- Modules are used to study linear algebra over rings.
- Read about modules.
Algebras
- An algebra is a vector space equipped with one or more operations.
- The operations must satisfy certain axioms, including associativity and distributivity.
- Learn more about algebras.
Applications
Algebraic structures find applications in various fields, including:
- Computer Science: Group theory is used in cryptography, while rings and fields are used in coding theory.
- Physics: Lie algebras are used in the study of symmetries in physics.
- Chemistry: Group theory is used to describe molecular structures.
Further Reading
Group Theory