# Algebra

[https://en.wikipedia.org/wiki/Algebra](https://en.wikipedia.org/wiki/Algebra)

**Algebra**, under a broad consideration, is the study of mathematical symbols and the rules for manipulating these symbols. Algebra, together with number theory, geometry and analysis forms a large part of math; it is used in many math branches because it studies the topics that frequently occur throughout mathematics. Algebra studies everything from elementary equation solving to the study of abstractions, such as groups, rings, fields.

**Elementary algebra** is similar to arithmetic, but it replaces concrete numbers with variables ranging over numbers in order to focus on the laws governing the manipulation of symbols in equations, rather then their immediate result.

**Abstract algebra**, the advance part of algebra, studies abstractions of mathematical objects into _algebraic structures_, such as a monoid, group, ring, field, etc.

**Boolean algebra**, introduced by George Boole in 1847, is a branch of algebra in which the values of the variables are the truth values, `True` and `False`.

**Linear algebra** is concerned with linear equations such as linear functions and their representations through matrices and vector spaces.

**Algebraic topology** uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to a homeomorphism, although usually most classify up to a homotopy equivalence.

**Universal algebra** (sometimes called _general algebra_) studies algebraic structures themselves, not examples/models of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as a single object of study. Universal algebra is the formalization of the general theory of algebraic structures. Different classes of non/algebraic objects, and relations between them, are studied in terms of the category theory.

## The various meanings of algebra

In math, the word "algebra" has several related meanings:

* Without an article: "algebra" denotes a part of math
*   With an article: "an algebra" or in the plural form, "algebras",  &#x20;

    it denotes a specific mathematical structure
*   With a qualifier (adjective), it denotes a part of algebra,  &#x20;

    such as "elementary algebra" or "abstract algebra"
*   With an article and a qualifier,   &#x20;

    it denotes an instance of an abstract structure:   &#x20;

    "a Lie algebra", "an associative algebra", "a vertex operator algebra"
*   a qualifier sometimes carries both meanings, as in this example:    &#x20;

    "Commutative algebra is the study of commutative rings, which are commutative algebras over the integers"
