Free algebra

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In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring (which may be regarded as a free commutative algebra).

Let R be a commutative ring. The free algebra on n indeterminates, {X₁, ..., X_n}, is the ring spanned by all linear combinations of products of the variables. This ring is denoted R<X₁, ..., X_n>. With the obvious scalar multiplication R<X₁, ..., X_n> 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.