Commutative Algebra

In this page, we will see how to perform basic operations in commutative algebra in Macaulay 2.

Ideal Manipulations

To create an ideal

S = QQ[x, y, z]
I = ideal (y - x^3, z - x^5, y*x^2 - z)

Groebner basis

GI = gb I
-- to see the generators of the Greobner basis
gens GI 

To compute the normal form of a polynomial $f$ with respect to an ideal $I$ (that is, the remainder of $f$ when divided by the Groebner basis of $I$), simply do the following:

f % I

Testing membership in the ideal $I$ is then done by:

f % I == 0