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Algebra may be divided roughly into the following categories:
* Elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra), also called second year and third year algebra;
* Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated; this includes, among other fields,
* Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
* Universal algebra, in which properties common to all algebraic structures are studied.
* Algebraic number theory, in which the properties of numbers are studied through algebraic systems. Number theory inspired much of the original abstraction in algebra.
* Algebraic geometry in its algebraic aspect.
* Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a geometric structure (a metric or a topology) which is compatible with the algebraic structure.
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