Definition:Algebra
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Disambiguation
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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.
- Boolean Algebra: A subfield of logic.
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:
- 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.
- Algebra is also the stage name of a contemporary R & B performer.
