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Free AlgebraIn abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. Let R be a ring. The free algebra on n indeterminates, X1, ..., Xn, is the ring spanned by all linear combinations of products of the variables. This ring is denoted R<X1, ..., Xn> Unlike in a polynomial ring, the variables do not commute. For example X1X2 does not equal X2X1. 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.)
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