4. Relating the Word and Ideal Membership Problems in Groups and Free Group Rings

In this section we want to point out how the Gröbner basis
methods as introduced in [MaRe93,Re95] for general monoid rings
when applied to group rings are related to
the word problem.
First we state that similar to theorem 1
the word problem for groups is equivalent to a restricted version of the
membership problem for ideals in a
free group ring.
Let the group be presented by a string rewriting system
such that
there exists an
involution
,
i.e for all
we have
,
,
and
the
.
Every group has such a presentation.
Notice that the set of rules T_{I} is confluent with respect to any admissible ordering
on .
By
we will denote the free group with presentation
.
The elements of
will be represented by freely reduced words, i.e. we assume
that the words do not contain any subwords of the form
.

Theorem 6 ([Re95,MaRe95])
Let
be a finite string rewriting system presenting a group
and without loss of generality for all
we assume that l and r are free reduced words.
We associate the set of polynomials
in
with T.
Then for
the following statements are equivalent: