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 R⟨X1,...,Xn⟩. 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 Xi), 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 Xi.
Read more about this topic: Free Algebra
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