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**.

## Definition

For *R* a commutative ring, the free (associative, unital) algebra on *n* indeterminates {*X*_{1},...,*X _{n}*} is the free

*R*-module with a basis consisting of all words over the alphabet {

*X*

_{1},...,

*X*} (including the empty word, which is the unit of the free algebra). This

_{n}*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 *R*-module elements is thus uniquely determined (because the multiplication in an *R*-algebra must be *R*-bilinear). This *R*-algebra is denoted *R*⟨*X*_{1},...,*X _{n}*⟩. This construction can easily be generalized to an arbitrary set

*X*of indeterminates.

In short, for an arbitrary set , the **free (associative, unital) R-algebra on X** is

with the *R*-bilinear multiplication that is concatenation on words, where *X** denotes the free monoid on *X* (i.e. words on the letters *X*_{i}), denotes the external direct sum, and *Rw* denotes the free *R*-module on 1 element, the word *w*.

For example, in *R*⟨*X*_{1},*X*_{2},*X*_{3},*X*_{4}⟩, for scalars *α, β, γ, δ* ∈ *R*, a concrete example of a product of two elements is

.

The non-commutative polynomial ring may be identified with the monoid ring over *R* of the free monoid of all finite words in the *X*_{i}.

## Contrast with Polynomials

Since the words over the alphabet {*X*_{1}, ...,*X _{n}*} form a basis of

*R*⟨

*X*

_{1},...,

*X*⟩, it is clear that any element of

_{n}*R*⟨

*X*

_{1}, ...,

*X*⟩ can be written uniquely in the form:

_{n}where are elements of *R* and all but finitely many of these elements are zero. This explains why the elements of *R*⟨*X*_{1},...,*X _{n}*⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates")

*X*

_{1},...,

*X*; the elements are said to be "coefficients" of these polynomials, and the

_{n}*R*-algebra

*R*⟨

*X*

_{1},...,

*X*⟩ is called the "non-commutative polynomial algebra over

_{n}*R*in

*n*indeterminates". Note that unlike in an actual 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 **Z**-algebras, 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 *n*-dimensional 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 *R*-algebras to the category of sets.

Free algebras over division rings are free ideal rings.

## See also

## References

- Berstel, Jean; Reutenauer, Christophe (2011).
*Noncommutative rational series with applications*. Encyclopedia of Mathematics and Its Applications.**137**. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007. - L.A. Bokut' (2001) [1994], "Free associative algebra",
*Encyclopedia of Mathematics*, EMS Press