## Buchberger Theory for Effective Associative Rings

Speaker: Prof. Teo Mora
Date: 09/03/2015
Time: 13:30 - 14:30

Location: RISC Seminar room

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