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 (which may be regarded as a free commutative algebra).

Definition

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1,...,Xn} is the free R-module with a basis consisting of all words over the alphabet {X1,...,Xn} (including the empty word, which is the unity 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:

and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted RX1,...,Xn⟩. 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

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

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