4. Conclusions

So far we have derived a generating set of solutions for a system of inhomogeneous equations using prefix Gröbner bases. Now in many monoid or group rings, e.g. when the respective structure is commutative, such prefix Gröbner bases cannot be expected to be finite in general. On the other hand there are examples where finite Gröbner bases exist for finitely generated one or even two-sided ideals in monoid or group rings with respect to reduction relations other than prefix reduction. In general monoid and group rings (one and two-sided) Gröbner bases can be characterized in terms of two kinds of test sets

• ${\sf SPOL}(F)$ corresponding to critical polynomials of the generating set $F$ called s-, g-, or m-polynomials depending on the context,
• ${\sf SAT}(F)$ corresponding to critical polynomial multiples of the generating set $F$ (compare the saturation techniques described e.g. in [5]).

Both sets are dependent on the reduction relation chosen for the monoid or group ring. Now for those cases where a set $G$ is a Gröbner basis and these two test sets ${\sf SPOL}(G)$ and ${\sf SAT}(G)$ are finite, a finite set of solutions for the homogeneous equation $g_1 \ast X_1 + \ldots + g_s \ast X_s = 0$ where $G = \{ g_1, \ldots, g_s \}$ can be derived as described in Section 2. Examples where these ideas are applicable since the test sets are finite and finite respective one-sided Gröbner bases exist are free and plain group rings. Even two-sided finite Gröbner bases and finite test sets exist for commutative monoid rings and polycyclic group rings. A similar extension can be done to treat systems of inhomogeneous equations as described in Section 3. The details are subject of an extended version of this paper.