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Let ''R'' be a Commutative Ring . The free algebra on ''n'' Indeterminate s, {''X1'', ..., ''Xn''}, is the ring spanned by all Linear Combination s of products of the variables. This ring is denoted ''R''<''X1'', ..., ''Xn''>. 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 Functor ial 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 . SEE ALSO |
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