Non-commutative Gröbner bases for left, right and two sided ideals in several classes of rings have been studied during the last years. Examples of such are non-communitative algebras over fields (Mora, [6]) and solvable polynomial rings (Kandry-Rody and Weispfenning, [3], Kredel, [4]). For the latter class the left, right and two sided ideal membership problem can be solved algorithmically by Gröbner basis methods.
One problem concerning this rings is that Dickson's lemma does not hold, i.e. there is, in general, an infinite sequence of terms where no term is divided (on the left resp. on the right) by any term occuring earlier in the sequence. This implies termination of a Buchberger algorithm-if given at all-must be proven by different means.
A new feature occurs in the class of rings subject of the present paper, a special case of which has been considered in [11] by Weispfenning and by the author in [9]. Though left and right ideals are not finitely generated in general there is a finite Gröbner basis for any finitely generated left and right ideal provided a special term order is used. This Gröbner bases can be computed by a Buchberger algorithm.
For other admissible term orders only infinite Gröbner bases exist in general even for finitely generated ideals and will be enumerated by a Buchberger algorithm. If (for a special case) a finite Gröbner basis exists the algorithm will terminate and return such a basis.
In general for iterated Ore extensions (with more than one iteration) there are no finite left or right Gröbner bases wrt. any given admissible term order.
The plan of the paper is as follows. The second section gives the basic properties of Ore extensions. We consider polynomial rings with an added substitution operator in the third section. It is shown that no finite Gröbner bases exist in general for most term orders, but if one exists, it can be computed.
A modification of Dickson's lemma, which allows to prove termination of a Buchberger algorithm wrt. a special term order is proven in section four.
Using the results of the preceding sections we show in the fifth section that finite left and right Gröbner bases in more general Ore textensions can be computed by a Buchberger algorithm.
Subject of the sixth section is the general non-existence of finite left and right Gröbner bases in iterated Ore extensions for any given admissible term order.
Ulrike Peiker, Martin Griebl