List of abstract algebra topics

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.

Basic language

Algebraic structures are defined primarily as sets with operations.

Algebraic structure

Subobjects: subgroup, subring, subalgebra, submodule etc.

Binary operation

Closure of an operation

Associative property

Distributive property

Commutative property

Unary operator

Additive inverse, multiplicative inverse, inverse element

Identity element

Cancellation property

Finitary operation

Arity

Structure preserving maps called homomorphisms are vital in the study of algebraic objects.

Homomorphisms

Kernels and cokernels

Image and coimage

Epimorphisms and monomorphisms

Isomorphisms

Isomorphism theorems

There are several basic ways to combine algebraic objects of the same type to produce a third object of the same type. These constructions are used throughout algebra.

Direct sum

Direct limit

Direct product

Inverse limit

Quotient objects: quotient group, quotient ring, quotient module etc.