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7.6.1 Free associative algebras

Let $V$ be a $K$-vector space, spanned by the symbols $x_1$,..., $x_n$. A free associative algebra in $x_1$,..., $x_n$ over $K$, denoted by $K$< $x_1$,..., $x_n$ > is also known as a tensor algebra $T(V)$ of $V$. It is an infinite dimensional $K$-vector space, where one can take as a basis the elements of the free monoid < $x_1$,..., $x_n$ >, identifying the identity element (the empty word) with the $1$ in $K$. In other words, the monomials of $K$ < $x_1$,..., $x_n$ > are the words of finite length in the finite alphabet { $x_1$,..., $x_n$ }. The algebra $K$< $x_1$,..., $x_n$ > is an integral domain, which is not Noetherian for $n>1$ (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 $K$< $x_1$,..., $x_n$ > modulo a two-sided ideal.