Characterization of relative Gröbner bases

We present a characterization of relative Gröbner bases and provide a result on the preservation of a relative Gröbner basis under changes of the involved orderings. Furthermore, we show that computing a relative Gröbner basis amounts to determining a finite basis of an ideal in a possibly non-Noetherian ring. We also provide an example where the method suggested by Zhou and Winkler for computing a relative Gröbner basis does not terminate.