Contrast With Polynomials
Since the words over the alphabet {X1, ...,Xn} form a basis of R⟨X1,...,Xn⟩, it is clear that any element of R⟨X1, ...,Xn⟩ can be uniquely written in the form:
where are elements of R and all but finitely many of these elements are zero. This explains why the elements of R⟨X1,...,Xn⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") X1,...,Xn; the elements are said to be "coefficients" of these polynomials, and the R-algebra R⟨X1,...,Xn⟩ is called the "non-commutative polynomial algebra over R in n indeterminates". Note that unlike in an actual polynomial ring, the variables do not commute. For example X1X2 does not equal X2X1.
More generally, one can construct the free algebra R⟨E⟩ on any set E of 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, 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 generators.
The construction of the free algebra on E is functorial 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 rings are free ideal rings.
Read more about this topic: Free Algebra
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