Definition:Algebra (Abstract Algebra)
From ProofWiki
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.
The above needs further explanation.
You can help ProofWiki by explaining it.
To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
If you would welcome a second opinion as to whether your explanation is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
