# 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.

## 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$_n$} (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:
    $\left(X_{i_{1}}X_{i_{2}}\cdots X_{i_{l}}\right)\cdot \left(X_{j_{1}}X_{j_{2}}\cdots X_{j_{m}}\right)=X_{i_{1}}X_{i_{2}}\cdots X_{i_{l}}X_{j_{1}}X_{j_{2}}\cdots X_{j_{m}},$
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 $X=\{X_{i}\,;\;i\in I\}$, the free (associative, unital) R-algebra on X is
    $R\langle X\rangle :=\bigoplus _{w\in X^{\ast }}Rw$
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$_i$), $\oplus$ 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
    $(\alpha X_{1}X_{2}^{2}+\beta X_{2}X_{3})\cdot (\gamma X_{2}X_{1}+\delta X_{1}^{4}X_{4})=\alpha \gamma X_{1}X_{2}^{3}X_{1}+\alpha \delta X_{1}X_{2}^{2}X_{1}^{4}X_{4}+\beta \gamma X_{2}X_{3}X_{2}X_{1}+\beta \delta X_{2}X_{3}X_{1}^{4}X_{4}$.
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$_n$⟩, it is clear that any element of R⟨X$_1$, ...,X$_n$⟩ can be written uniquely in the form:
    $\sum \limits _{k=0}^{\infty }\,\,\,\sum \limits _{i_{1},i_{2},\cdots ,i_{k}\in \left\lbrace 1,2,\cdots ,n\right\rbrace }a_{i_{1},i_{2},\cdots ,i_{k}}X_{i_{1}}X_{i_{2}}\cdots X_{i_{k}},$
where $a_{i_{1},i_{2},...,i_{k}}$ 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$_n$; the elements $a_{i_{1},i_{2},...,i_{k}}$ are said to be "coefficients" of these polynomials, and the R-algebra R⟨X$_1$,...,X$_n$⟩ is called the "non-commutative polynomial algebra over 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.