Free Algebra - Definition

Definition

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X₁,...,X_n} is the free R-module with a basis consisting of all words over the alphabet {X₁,...,X_n} (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 R⟨X₁,...,X_n⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.

In short, for an arbitrary set, 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 X_i), denotes the external direct sum, and Rw denotes the free R-module on 1 element, the word w.

For example, in R⟨X₁,X₂,X₃,X₄⟩, for scalars α,β,γ,δ ∈R, a concrete example of a product of two elements is .

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.