Risc Colloquium

The extension of Buchberger Theory and Algorithm from the classical case of
polynomial rings over a field[1, 2, 3] to the case of (non necessarily commutative) monoid rings over a (non necessarily free) monoid and a principal ideal ring was immediately performed by a series of milestone papers: Zacharias’ [9] approach to canonical forms, Spear’s[7] theorem which extends Buchberger Theory to each effectively given rings, M¨oller’s[5] reformulation of Buchberger Algorithm in terms of lifting.

Since the universal property of the free monoid ring Q := Z[Z*] over Z and the
monoid Z* of all words over the alphabet Z grants that each ring with identity
A can be presented as a quotient A = Q/I of a free monoid ring Q modulo a
bilateral ideal I in Q, in order to impose a Buchberger Theory over any effectiveassociative ring it is sufficient to reformulate it in filtration-valuation terms [8,4, 6] and apply the results quoted above; in particular Zacharias canonical forms allow to effectively present A and its elements, Spear’s theorem describes how Q imposes its natural filtration on A and a direct application of M¨oller’s lifting theorem to such filtration allows to characterize the required S-polynomials.

References:

[1] Buchberger B., Ein Algorithmus zum Auffinden der Basiselemente des
Restklassenringes nach einem nulldimensionalen Polynomideal, Ph. D.
Thesis, Innsbruck (1965)

[2] Buchberger B., Ein algorithmisches Kriterium f¨ur die L¨osbarkeit eines algebraischen Gleischunssystem, Aeq. Math. 4 (1970), 374–38

[3] Buchberger B., Gr¨obner Bases: An Algorithmic Method in Polynomial Ideal
Theory, in Bose N.K. (Ed.) Multidimensional Systems Theory (1985), 184–
232, Reider

[4] T. Mora, Seven variations on standard bases, (1988)
ftp://ftp.disi.unige.it/person/MoraF/PUBLICATIONS/7Varietions.tar.gz