A division algebra is a possibly non-associative algebra , typically over a field , with the property that for any then implies either or (or whenever ).
If is finite-dimensional (over a field ), this is equivalent to assuming that for any nonzero , the operations of left multiplication and right multiplication by are invertible. If furthermore is also associative and unital, this is also equivalent to the existence, for each nonzero , of an element with . However, it is easy to construct nonassociative unital finite-dimensional algebras over (the field of real numbers) such that either:
is not a division algebra but for each nonzero there exists with .
is a division algebra but there exists nonzero for which there is no with .