The free associative algebra 
on a set 
over a field 
is the vector space with basis
all finite words in 
,
with multiplication given by concatenation and extended bilinearly. Every map from
to an associative 
-algebra extends uniquely to an algebra homomorphism from 
.
Free Algebra
See also
Algebra, Free Group, Tensor Algebra, Universal PropertyExplore with Wolfram|Alpha
References
Cohn, P. M. Free Rings and Their Relations, 2nd ed. London, England: Academic Press, 1985.Cite this as:
Weisstein, Eric W. "Free Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FreeAlgebra.html