# Definition:Algebra (Abstract Algebra)

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## Definition

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.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**algebra**:**3.**