On noncommutative Gröbner bases over rings

Let R be a commutative ring. It is proved that for verification of whether a set of elements {f α} of the free associative algebra over R is a Gröbner basis (with respect to some admissible monomial order) of the (bilateral) ideal that the elements f α generate it is sufficient to check the reducibility to zero of S-polynomials with respect to {f α} iff R is an arithmetical ring. Some related open questions and examples are also discussed. © 2007 Springer Science+Business Media, Inc.