Editing Free algebra (section)

==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]], the variables do not [[commutative operation|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 [[generating set|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]] on an ''n''-dimensional [[vector space]].  For a more general coefficient ring, the same construction works if we take the [[free module]] on ''n'' [[generating set|generators]].

The construction of the free algebra on ''E'' is [[functor]]ial 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 ring]]s are [[free ideal ring]]s.