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division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible.
==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).
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