This article is about free algebras in ring theory. For the more general free algebras in universal algebra see free object.

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). Contents 1 Definition 2 Contrast with Polynomials 3 See also 4 References Definition

For R a commutative ring the free (associative unital) algebra on n indeterminates X1...Xn (where "indeterminate" is just a fancy word for "symbol" and the symbols X1...Xn are supposed to be pairwise distinct) 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. Contrast with Polynomials

Since the words over the alphabet X1 ...Xn form a basis of RX1...Xn it is clear that any element of RX1 ...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 RX1...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 RX1...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 RE 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 ZE.

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. See also Tensor algebra Free object Noncommutative ring References L.A. Bokut' (2001) "Free associative algebra" in Hazewinkel Michiel Encyclopaedia of Mathematics Springer ISBN 978-1556080104 http://eom.springer.de/f/f041520.htm