Free algebra

In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring.

Let R be a ring. The free algebra on n indeterminates, X₁, ..., X_n, is the ring spanned by all linear combinations of products of the variables. This ring is denoted R<X₁, ..., X_n>

Unlike in a polynomial ring, the variables do not commute. For example X₁X₂ does not equal X₂X₁.

Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. (For a more general coefficient ring, the same construction works if we take the free module on n generators.)