|Title||Left and right Gröbner bases in Ore extensions of polynomial rings|
|Author(s)|| Michael Pesch|
|Type||Technical Report, Misc|
|Abstract||We show, that Groebner bases can be computed for left and right ideals of certain Ore extensions of polynomial rings.|
Consider an Ore extension K[X_1, ... , X_m] [Y; alpha, gamma] of a polynomial ring over a field, where alpha (X_i) = X_i^(l_i) + f_i for some f_i in K[X_1, ... ,Xm], f_i ? X^(l_i)_i for some admissible term order ?, e_i in N ?? 0 ? and alpha | K = id. This rings are in general neither right nor left Noetherian. Nevertheless finite left and right Groebner bases for finitely generated left and right ideals do exist for special term orders. Using this Groebner bases the ideal membership problem can be solved. For other term orders no finite Groebner bases exist in general. Finite left and right Groebner bases can be computed using left/right reduction and s-polynomials (based on right/left divisibility and left/right least common multiples) in Buchbergers algorithm. Termination can be proven by a modified Dickson lemma, using the special term order.