Monomial ideal

In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.

be the polynomial ring over

The following three conditions are equivalent for an ideal

is a monomial ideal if it satisfies any of these equivalent conditions.

if and only if every monomial ideal term

[1] Proof: Suppose

can be written as a sum of multiples of the

will be a sum of multiples of monomial terms for at least one of the

The following illustrates an example of monomial and polynomial ideals.

is in I, since each term is a multiple of an element in J, i.e., they can be rewritten as

is not in J, since its terms are not multiples of elements in J. Bivariate monomial ideals can be interpreted as Young diagrams.

has a unique minimal generating set of

in the plane, the figure formed by the monomials in

In this figure, the minimal generators form the inner corners of a Young diagram.

lie below the staircase, and form a vector space basis of the quotient ring

The set of grid points

corresponds to the minimal monomial generators

Then as the figure shows, the pink Young diagram consists of the monomials that are not in

The points in the inner corners of the Young diagram, allow us to identify the minimal monomials

In general, to any set of grid points, we can associate a Young diagram, so that the monomial ideal is constructed by determining the inner corners that make up the staircase diagram; likewise, given a monomial ideal, we can make up the Young diagram by looking at the

and representing them as the inner corners of the Young diagram.

The coordinates of the inner corners would represent the powers of the minimal monomials in

Thus, monomial ideals can be described by Young diagrams of partitions.

has fixed points corresponding to monomial ideals only, which correspond to integer partitions of size n, which are identified by Young diagrams with n boxes.

By the monomial order, we can state the following definitions for a polynomial in

Definition[1] Note that

in general depends on the ordering used; for example, if we choose the lexicographical order on

In addition, monomials are present on Gröbner basis and to define the division algorithm for polynomials in several indeterminates.

Notice that for a monomial ideal

, the finite set of generators