Definition:Algebra (Abstract Algebra)

Definition
In the context of Abstract Algebra, in particular ring theory and linear algebra, the following varieties of algebra exist:


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


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


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


 * Division Algebra: an algebra over a field $\left({A_F, \oplus}\right)$ such that $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$.


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


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


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


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


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


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