Division algebra is a type of algebraic structure where division is possible, except by zero. More formally, a division algebra is a vector space over a field \( F \) equipped with a bilinear multiplication operation that satisfies the following conditions: 1. **Non-Associativity or Associativity**: In a general division algebra, multiplication can be either associative or non-associative. If it is associative, the algebra is called an associative division algebra.
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A division algebra over a field is a division ring equipped with a unital algebra structure over , with mapping into its center of an associative algebra. Every nonzero element is invertible; it need not be finite-dimensional over .
An algebra over a field where division exists.