In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring (which may be regarded as a free commutative algebra).
For R a commutative ring, the free (associative, unital) algebra on n indeterminates, {X_{1}, ..., X_{n}}, is the ring spanned by all sums of products of the variables. This ring is denoted R<X_{1}, ..., X_{n}>. With the obvious scalar multiplication R<X_{1}, ..., X_{n}> forms an algebra over R. Unlike in a polynomial ring, the variables do not commute. For example X_{1}X_{2} does not equal X_{2}X_{1}.
More generally, one can construct the free algebra R<E> on any set E of generators. Since rings may be regarded as Zalgebras, a free ring on E can be defined as the free algebra Z<E>.
Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an ndimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators.
The construction of the free algebra on E is functorial in nature and satisfies an appropriate universal property. The free algebra functor is left adjoint to the forgetful functor from the category of Ralgebras to the category of sets.
Free algebras over division rings are free ideal rings.
