Monomial ideal

In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.
A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.
Definitions and Properties.
Let formula_1 be a field and formula_2 be the polynomial ring over formula_1 with "n" variables formula_4. 
A monomial in formula_5 is a product formula_6 for an "n"-tuple formula_7 of nonnegative integers.
The following three conditions are equivalent for an ideal formula_8:
We say that formula_18 is a monomial ideal if it satisfies any of these equivalent conditions.
Given a monomial ideal formula_19, formula_20 is in formula_9 if and only if every monomial ideal term formula_22 of formula_23 is a multiple of one the formula_24.
Proof:
Suppose formula_25 and that formula_20 is in formula_9. Then formula_28, for some formula_29. 
For all formula_30, we can express each formula_22 as the sum of monomials, so that formula_23 can be written as a sum of multiples of the formula_33. Hence, formula_23 will be a sum of multiples of monomial terms for at least one of the formula_33.
Conversely, let formula_25 and let each monomial term in formula_37 be a multiple of one of the formula_33 in formula_9. Then each monomial term in formula_13 can be factored from each monomial in formula_23. Hence formula_23 is of the form formula_43 for some formula_44, as a result formula_45. 
The following illustrates an example of monomial and polynomial ideals.
Let formula_46 then the polynomial formula_47 is in , since each term is a multiple of an element in , i.e., they can be rewritten as formula_48 and formula_49 both in . However, if formula_50, then this polynomial formula_51 is not in , since its terms are not multiples of elements in .
Monomial Ideals and Young Diagrams.
A monomial ideal can be interpreted as a Young diagram. Suppose formula_52, then formula_9 can be interpreted in terms of the minimal monomials generators as formula_54, where formula_55 and formula_56. The minimal monomial generators of formula_9 can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram.
The monomials not in formula_9 lie inside the staircase, and these monomials form a vector space basis for the quotient ring formula_59.
Consider the following example. 
Let formula_60 be a monomial ideal. Then the set of grid points formula_61 corresponds to the minimal monomial generators formula_62 in formula_9. Then as the figure shows, the pink Young diagram consists of the monomials that are not in formula_9. The points in the inner corners of the Young diagram, allow us to identify the minimal monomials formula_65 in formula_9 as seen in the green boxes. Hence, formula_67.
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 formula_68 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 formula_13. Thus, monomial ideals can be described by Young diagrams of partitions.
Moreover, the formula_70-action on the set of formula_71 such that formula_72 as a vector space over formula_73 has fixed points corresponding to monomial ideals only, which correspond to partitions of size "n", which are identified by Young diagrams with "n" boxes.
Monomial Ordering and Gröbner Basis.
A monomial ordering is a well ordering formula_74 on the set of monomials such that if formula_75 are monomials, then formula_76.
By the monomial order, we can state the following definitions for a polynomial in formula_77. 
"Definition"
Note that formula_92 in general depends on the ordering used; for example, if we choose the lexicographical order on formula_93 subject to "x" > "y", then formula_94, but if we take "y" > "x" then formula_95.
In addition, monomials are present on Gröbner basis and to define the division algorithm for polynomials with several variables. 
Notice that for a monomial ideal formula_96, the finite set of generators formula_97 is a Gröbner basis for formula_9. To see this, note that any polynomial formula_45 can be expressed as formula_100 for formula_101. Then the leading term of formula_23 is a multiple for some formula_103. As a result, formula_92 is generated by the formula_103 likewise.