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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).

For R a commutative ring, the free (associative, unital) algebra on n indeterminates, {X1, ..., Xn}, is the ring spanned by all sums of products of the variables. This ring is denoted R<X1, ..., Xn>. With the obvious scalar multiplication R<X1, ..., Xn> forms an algebra over R. Unlike in a 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.

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