Any commutative unital domain can be embedded into a smallest field in a unique way, its field of fractions. For noncommutative domains, both the existence and uniqueness (of a smallest skew field) do not need to hold, and are difficult to decide.
A classical theorem guarantees the existence for a class of Ore domain?s that is the domains for which the subset of regular elements is an Ore subset?; then the corresponding Ore localization is a skewfield and the localization map is an injection. Another important case is the case of free associative algebras; in that case the solution is not unique and the first embedding was constructed by Amitsur, and by a different procedure by P. M. Cohn. This skewfield is called the free field. It has been more recently used in formulating the theory of quasideterminants.