Monomial Ideal

A monomial ideal in a polynomial ring over a field is an ideal generated by monomials. Here monomials have coefficient 1. Membership of a monomial is equivalent to divisibility by one of the generating monomials. The monomials outside the monomial ideal give a vector basis of the quotient ring as a vector space over the field (Bruns and Herzog 1998).

For example, the ideal has quotient vector basis , so the Hilbert series of is . Every monomial ideal is a homogeneous ideal for the grading by total degree. The lex-plus-powers ideal imposes a particular order condition on its monomials.