
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 twosided 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 twosided ideal.
