and 
) and then the computation of this S-polynomial's normal
form (say r) using other polynomials
(say, 
).
One of the side effects of this type of computation is
the fact that s lies in the ideal generated by
. If this ideal membership
relationship is
stored during the process of running the GBA and
one wants to find a smaller generating set for the output of
the GBA,
then it is helpful
to use these ideal membership relationships.