# Definition:Algebra

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Algebra** may refer to:

- Algebra: The field of mathematics concerned with manipulation of equations.
- Abstract Algebra: The subfield of algebra concerned with abstract algebraic structures.
- Linear Algebra: The subfield of abstract algebra concerned with modules.
- Vector Algebra: The subfield of linear algebra concerned with vectors.

**Algebra** may also refer to structures in various branches of mathematics, for example:

### Algebra (Abstract Algebra)

In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of **algebra** exist:

- Definition:Algebra over Ring: an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus: G^2 \to G$.

- Definition:Algebra over Field: a vector space $G_F$ over a field $F$ with a bilinear mapping $\oplus: G^2 \to G$.

- Definition:Real Algebra: an algebra over a field where the field in question is the field of real numbers $\R$.

- Definition:Division Algebra: an algebra over a field $\struct {A_F, \oplus}$ such that $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$.

- Definition:Associative Algebra: an algebra over a ring in which the bilinear mapping $\oplus$ is associative.

- Definition:Unitary Algebra, also known as a Unital Algebra: an algebra over a ring $\struct {A_R, \oplus}$ in which there exists an identity element, that is, a
**unit**, usually denoted $1$, for $\oplus$.

- Definition:Unitary Division Algebra: a division algebra $\struct {A_F, \oplus}$ in which there exists an identity element, that is, a
**unit**, usually denoted $1$, for $\oplus$.

- Definition:Graded Algebra: an algebra over a ring where the ring has a gradation, that is, is a graded ring.

- Definition:Filtered Algebra: an algebra over a field which has a sequence of subalgebras which constitute a gradation.

- Definition:Quadratic Algebra: a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2.

- Algebra is also the stage name of a contemporary R & B performer.

## Linguistic Note

The word **algebra** originates from the Arabic word **al-ğabr**, meaning **balancing**, **reduction** or **restoration**.

It originates from the name of a book, circa $825$ C.E., by Muhammad ibn Musa al-Khwarizmi:

*Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala*(*The Compendious Book on Calculation by Completion and Balancing*)