  We investigate the reduction of Feynman integrals to master integrals using
Gr\"obner bases in a rational double-shift algebra Y in which the
integration-by-parts (IBP) relations form a left ideal. The problem of reducing
a given family of integrals to master integrals can then be solved once and for
all by computing the Gr\"obner basis of the left ideal formed by the IBP
relations. We demonstrate this explicitly for several examples. We introduce
so-called first-order normal-form IBP relations which we obtain by reducing the
shift operators in Y modulo the Gr\"obner basis of the left ideal of IBP
relations. For more complicated cases, where the Gr\"obner basis is
computationally expensive, we develop an ansatz based on linear algebra over a
function field to obtain the normal-form IBP relations.
