Free algebra

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X₁,...,X_n} is the free R-module with a basis consisting of all words over the alphabet {X₁,...,X_n} (including the empty word, which is the unit of the free algebra). This R-module becomes an R-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words:

and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted R⟨X₁,...,X_n⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.

In short, for an arbitrary set ${\displaystyle X=\{X_{i}\,;\;i\in I\}}$, the free (associative, unital) R-algebra on X is

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