On noncommutative Gröbner bases over rings

Let $R$ be a commutative ring. It is proved that for verification whether a set of elements $\{f_\alpha\}$ 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_\alpha $ generate it is sufficient to check reducibility to zero of $S$-polynomials with respect to $\{f_\alpha\}$ iff $R$ is an arithmetical ring. Some related open questions and examples are also discussed.