the F4 Algorithm which does not use criteria for unnecessary obstructions. For an overview on rules which have been developed see for instance [@Co07].

In this paper, we present generalizations of the Gebauer-Möller criteria for non-commutative polynomials. They cover not only the known cases of useless obstructions discussed in [@Mo94], Lemma 5.11 and [@Co07], but form a complete analogue of the results in the commutative case. One of the key ingredients we use for this purpose is the consideration of obstructions without overlaps. We detect useless obstructions, i.e. obstructions that can be represented by other obstructions, using not only obstructions with overlaps but using also those without overlaps. We show that the consideration of obstructions without overlaps does not increase unnecessary computations, since a Gröbner representation is inherent in the S-polynomial of every obstruction without overlaps. Consequently, we reduce the number of obstructions efficiently and obtain a non-commutative version of the Gebauer-Möller criteria.

This paper is organised as follows. In Section \[sec2\] we recall the basic theory of Gröbner bases for two-sided ideals in non-commutative polynomial rings. In particular, we introduce and study obstructions (see Definitions \[sec2def4\] and \[sec2def9\], and Lemmas \[sec2lem6\] and \[sec2lem8\]), present the Buchberger Criterion (see Proposition 