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{{Short description|Free object in the category of associative algebras}}
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In [[Mathematics|mathematics]], especially in the area of [[Abstract algebra|abstract algebra]] known as [[Ring theory|ring theory]], a '''free algebra''' is the noncommutative analogue of a [[Polynomial ring|polynomial ring]] since its elements may be described as "polynomials" with non-commuting variables. Likewise, the [[Polynomial ring|polynomial ring]] may be regarded as a '''free commutative algebra'''.

==Definition==
For ''R'' a [[Commutative ring|commutative ring]], the free (associative, unital) [[Algebra (ring theory)|algebra]] on ''n'' [[Indeterminate (variable)|indeterminate]]s {''X''<sub>1</sub>,...,''X<sub>n</sub>''} is the [[Free module|free ''R''-module]] with a basis consisting of all words over the alphabet {''X''<sub>1</sub>,...,''X<sub>n</sub>''} (including the empty word, which is the unit of the free algebra). This ''R''-module becomes an [[Algebra (ring theory)|''R''-algebra]] by defining a multiplication as follows: the product of two basis elements is the [[Concatenation|concatenation]] of the corresponding words:

:<math>\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},</math>

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''<sub>1</sub>,...,''X<sub>n</sub>''⟩. This construction can easily be generalized to an arbitrary set ''X'' of indeterminates.

In short, for an arbitrary set <math>X=\{X_i\,;\; i\in I\}</math>, the '''free (associative, unital) ''R''-[[Algebra (ring theory)|algebra]] on ''X''''' is 
:<math>R\langle X\rangle:=\bigoplus_{w\in X^\ast}R w</math>
with the ''R''-bilinear multiplication that is concatenation on words, where ''X''* denotes the [[Free monoid|free monoid]] on ''X'' (i.e. words on the letters ''X''<sub>i</sub>), <math>\oplus</math> denotes the external [[Direct sum of modules|direct sum]], and ''Rw'' denotes the [[Free module|free ''R''-module]] on 1 element, the word ''w''.

For example, in ''R''⟨''X''<sub>1</sub>,''X''<sub>2</sub>,''X''<sub>3</sub>,''X''<sub>4</sub>⟩, for scalars ''α, β, γ, δ'' ∈ ''R'', a concrete example of a product of two elements is 

<math>(\alpha X_1X_2^2 + \beta X_2X_3)\cdot(\gamma X_2X_1 + \delta X_1^4X_4) = \alpha\gamma X_1X_2^3X_1 + \alpha\delta X_1X_2^2X_1^4X_4 + \beta\gamma X_2X_3X_2X_1 + \beta\delta X_2X_3X_1^4X_4</math>.

The non-commutative polynomial ring may be identified with the [[Monoid ring|monoid ring]] over ''R'' of the [[Free monoid|free monoid]] of all finite words in the ''X''<sub>''i''</sub>.

==Contrast with polynomials==
Since the words over the alphabet {''X''<sub>1</sub>, ...,''X<sub>n</sub>''} form a basis of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩, it is clear that any element of ''R''⟨''X''<sub>1</sub>, ...,''X<sub>n</sub>''⟩ can be written uniquely in the form:

:<math>\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},</math>

where <math>a_{i_1,i_2,...,i_k}</math> are elements of ''R'' and all but finitely many of these elements are zero. This explains why the elements of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") ''X''<sub>1</sub>,...,''X<sub>n</sub>''; the elements <math> a_{i_1,i_2,...,i_k}</math> are said to be "coefficients" of these polynomials, and the ''R''-algebra ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ is called the "non-commutative polynomial algebra over ''R'' in ''n'' indeterminates". Note that unlike in an actual [[Polynomial ring|polynomial ring]], the variables do not commute. For example, ''X''<sub>1</sub>''X''<sub>2</sub> does not equal ''X''<sub>2</sub>''X''<sub>1</sub>.

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 (mathematics)|field]], the free algebra on ''n'' indeterminates can be constructed as the [[Tensor algebra|tensor algebra]] on an ''n''-dimensional [[Vector space|vector space]].  For a more general coefficient ring, the same construction works if we take the [[Free module|free module]] on ''n'' generators.

The construction of the free algebra on ''E'' is [[Functor|functor]]ial in nature and satisfies an appropriate [[Universal property|universal property]]. The free algebra functor is left adjoint to the [[Forgetful functor|forgetful functor]] from the category of ''R''-algebras to the [[Category of sets|category of sets]].

Free algebras over [[Division ring|division ring]]s are [[Free ideal ring|free ideal ring]]s.

==See also==

*Cofree coalgebra
*[[Tensor algebra]]
*[[Free object]]
*[[Noncommutative ring]]
*[[Rational series]]

==References==
* {{cite book | last1=Berstel | first1=Jean | last2=Reutenauer | first2=Christophe | title=Noncommutative rational series with applications | series=Encyclopedia of Mathematics and Its Applications | volume=137 | location=Cambridge | publisher={{wipe|Cambridge University Press}} | year=2011 | isbn=978-0-521-19022-0 | zbl=1250.68007 }}
* {{springer|id=f/f041520|author=L.A. Bokut'|title=Free associative algebra}}

[[Category:Algebras]]
[[Category:Ring theory]]
[[Category:Free algebraic structures]]

{{Sourceattribution|Free algebra}}