Category:Definitions/Division Algebras
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This category contains definitions related to Division Algebras.
Related results can be found in Category:Division Algebras.
Let $\struct {A_F, \oplus}$ be an algebra over field $F$ such that $A_F$ does not consist solely of the zero vector $\mathbf 0_A$ of $A_F$.
Definition 1
$\struct {A_F, \oplus}$ is a division algebra if and only if:
- $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$
Definition 2
$\struct {A_F, \oplus}$ is a division algebra if and only if it has no zero divisors:
- $\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$
Subcategories
This category has the following 2 subcategories, out of 2 total.
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U
Pages in category "Definitions/Division Algebras"
The following 5 pages are in this category, out of 5 total.