Editing Free algebra (section)

==Definition==
For ''R'' a [[commutative ring]], the free ([[associative]], [[unital algebra|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 [[Word (mathematics)|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]] 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 algebra|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]] 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]] over ''R'' of the [[free monoid]] of all finite words in the ''X''<sub>''i''</sub>.