  Classically, Groebner bases are computed by first prescribing a set monomial
order. Moss Sweedler suggested an alternative and developed a framework to
perform such computations by using valuation rings in place of monomial orders.
We build on these ideas by providing a class of valuations on rational function
fields of two variables that are suitable for this framework. For these
valuations, we explicitly compute the image of an underlying polynomial ring
and use this to perform computations concerning ideals in the polynomial ring.
Interestingly, for these valuations, some ideals have a finite Groebner basis
with respect to the valuation that is not a Groebner basis with respect to any
monomial order, whereas other ideals only have Groebner bases that are infinite
with respect to the valuation.
