Mike, this answer appears wrong, because of the appearance of the extra constant:

i43 : R = (QQ[w]/(w^4+w^3+w^2+w+1))[x,y];

i44 : newMonomialIdeal(R,rawMonomialIdeal(raw matrix {{x}}, 0))

                           3    2
o44 = monomialIdeal (x, - w  - w  - w - 1)

o44 : MonomialIdeal of R
