f % N
pseudoRemainder(f,N)
This method gives a randomized algorithm for ideal membership. If $f$ lies in the saturated ideal of each of the chains of the network, then the output is always zero. Otherwise, it returns a nonzero element with high probability.
As an example, consider the ideal of cyclically adjacent minors.







It is assumed that the base field has sufficiently many elements. For small finite fields one must work over a suitable field extension.