nLab
division algebra

A division algebra is a possibly non-associative algebra $A$, typically over a field $k$, with the property that $ab=0$ implies either $a=0$ or $b=0$ (or $ab\ne 0$ whenever $a,b\ne 0$). If $A$ is finite-dimensional (over a field), this is equivalent to assuming that for any nonzero $a\in A$, the operations of left multiplication and right multiplication by $a$ are invertible. If furthermore $A$ is also associative and unital, this is also equivalent to the existence, for each nonzero $a\in A$, of an element $a^{-1}\in A$ with $a{a}^{-1}=a^{-1}a=1$. However, it is easy to construct nonassociative unital finite-dimensional algebras over $ℝ$ (the field of real numbers) such that either:

• $A$ is not a division algebra but for each nonzero $a\in A$ there exists $a^{-1}\in A$ with $a{a}^{-1}=a^{-1}a=1$.

• $A$ is a division algebra but there exists nonzero $a\in A$ for which there is no $a^{-1}\in A$ with $a{a}^{-1}=a^{-1}a=1$.

For details see:

• John Baez, The octonions, section 1.1: Preliminaries, Bull. Amer. Math. Soc. 39 (2002), 145-205. (web) (website version)

For applications to physics, the most interesting division algebras are probably the normed division algebras: the real numbers, complex numbers, quaternions and octonions. These have important relations to supersymmetry.