### 7.6.1 Free associative algebras

Let be a -vector space, spanned by the symbols ,..., . A free associative algebra in ,..., over , denoted by < ,..., > is also known as a tensor algebra of . It is an infinite dimensional -vector space, where one can take as a basis the elements of the free monoid < ,..., >, identifying the identity element (the empty word) with the in . In other words, the monomials of < ,..., > are the words of finite length in the finite alphabet { ,..., }. The algebra < ,..., > is an integral domain, which is not Noetherian for (hence, a two-sided Groebner basis of a finitely generated ideal might be infinite). The free associative algebra can be regarded as a graded algebra in a natural way.

Any finitely presented associative algebra is isomorphic to a quotient of < ,..., > modulo a two-sided ideal.

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