\begin{definition}[Definition:Algebra (Abstract Algebra)]
In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of '''algebra''' exist:
* Definition:Boolean Algebra
* 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 $\left({A_F, \oplus}\right)$ such that $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$.
* 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 $\left({A_R, \oplus}\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\oplus$.
* Definition: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 $\oplus$.
* 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.
* Definition:Lie Algebra
\end{definition}