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# 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 TI 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:
(1)
.
(2)
.

Proof : 1.11.1

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