Let S be a polynomial ring, and let R be the subring generated by
a subset of the variables of S. One important construction is to
compute generators for 
, where I is an ideal of S: i.e.
eliminate the variables not in the subring R.
Let ;SPMgt; be any monomial order which satisfies the following property:
where 
is the lead monomial of f in this monomial order.
We can then compute via the following simple but important
fact.
The most important monomial orders which satisfy the above condition are the elimination and product orders defined in section 5.4. For example, let S = k[s,t,a,b,c,d], and let the monomial order ;SPMgt; be the product order s,t ;SPMgt;;SPMgt; a,b,c,d. The ring command to produce the ring with this order is
% ring S ! characteristic (if not 31991) ? <return> ! number of variables ? 6 ! 6 variables, please ? stabcd ! variable weights (if not all 1) ? 1 1 3 ! monomial order (if not rev. lex.) ? 2 4 ; largest degree of a monomial : 512 512
Let 
. We use the above proposition to compute
for any ideal I of S. Several examples are given in
Appendix A.
Macaulay stores each monomial in the above ring S as a list of three numbers: the part of the monomial in s,t, the part in the variables a,b,c,d, and finally the module component number (see section 5.4 for more details). The ring R is generated by those monomials of S whose ``value'' on the first ``block'' of variables is zero, i.e. a,b,c,d. More generally, we can define the subring generated by the variables (equivalently, monomials) whose value on the first n blocks is zero:
In the above example, 
, and 
. These subrings
are always generated by a subset of the variables. If the monomial order
has been chosen to be an elimination order or a product order, the subring
of most interest is 
, the subring generated by the variables whose value
on the first block is zero.
This notation is used in the descriptions below.