Definitions
Formally, we start with an algebra D over a field, and assume that D does not just consist of its zero element. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb.
For associative algebras, the definition can be simplified as follows: an associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1≠0 and every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1).
Read more about this topic: Division Algebra
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