Free algebra

Summary

The free algebra $k\langle X \rangle$ on a set $X$ over a commutative ring $k$ is the algebra of polynomials in noncommuting variables from $X$. It is the free object among associative algebras over $k$: any map from $X$ into an algebra extends uniquely to an algebra homomorphism. So every algebra generated by a set $X$ is a quotient of $k\langle X \rangle$, and algebras are often given by generators and relations in this way. It is the noncommutative analogue of the polynomial ring and is isomorphic to the tensor algebra of the free module with basis $X$.