comparison of ideals

edit

To find out, you can look in the __cmp__ special method of the object you're interested in. This function returns negative if x<y, zero if x==y, and positive if x>y. Looking in __cmp__ interactively is probably a better idea than browsing the source, because if you do that you may guess wrong about which __cmp__ is being used (like I did!).

For example:

sage: P.<x> = PolynomialRing(QQ)
sage: I = Ideal([x^2-2*x+1, x^2-1])
sage: J = Ideal([4 + 3*x + x^2, 1 + x^2])
sage: I.__cmp__??
Type:       instancemethod
Base Class: <type 'instancemethod'>
String Form:    <bound method Ideal_1poly_field.__cmp__ of Principal ideal (x - 1) of Univariate Polynomial Ring in x over Rational Field>
Namespace:  Interactive
File:       /Applications/sage/local/lib/python2.6/site-packages/sage/rings/ideal.py
Definition: I.__cmp__(self, other) 
Source:
    def __cmp__(self, other):
        if not isinstance(other, Ideal_generic):
            other = self.ring().ideal(other)

    if not other.is_principal():
        return -1

    if self.is_zero():
        if not other.is_zero():
            return -1
        return 0

    # is other.gen() / self.gen() a unit in the base ring?
    g0 = other.gen()
    g1 = self.gen()
    if g0.divides(g1) and g1.divides(g0):
        return 0
    return 1
sage: I.gen(), J.gen()
(x - 1, 1)
sage: I == J, I <= J, I >= J
(False, False, True)

sage: # or exactly the same case but now (potentially) multivariate
sage: P.<x,y> = PolynomialRing(QQ,2)
sage: I = Ideal([x^2-2*x+1, x^2-1])
sage: J = Ideal([4 + 3*x + x^2, 1 + x^2])
sage: I.__cmp__??
[docstring removed]
    # first check the type
    if not isinstance(other, MPolynomialIdeal):
        return 1

    # the ideals may be defined w.r.t. to different term orders
    # but are still the same.
    R = self.ring()
    S = other.ring()
    if R is not S: # rings are unique
        if type(R) == type(S) and (R.base_ring() == S.base_ring()) and (R.ngens() == S.ngens()):
            other = other.change_ring(R)
        else:
            return cmp((type(R), R.base_ring(), R.ngens()), (type(S), S.base_ring(), S.ngens()))

    # now, check whether the GBs are cached already
    if self.groebner_basis.is_in_cache() and other.groebner_basis.is_in_cache():
        l = self.groebner_basis()
        r = other.groebner_basis()
    else: # use easy GB otherwise
        try:
            l = self.change_ring(R.change_ring(order="degrevlex")).groebner_basis()
            r = other.change_ring(R.change_ring(order="degrevlex")).groebner_basis()
        except AttributeError: # e.g. quotient rings
            l = self.groebner_basis()
            r = other.groebner_basis()
    return cmp(l,r)

sage: I.groebner_basis(), J.groebner_basis()
([x - 1], [1])

sage: I == J, I <= J, I >= J (False, False, True)

So it seems (modulo some details) it's basically comparing the Sequences of the groebner bases, and that appears to fall through to simply sorting the underlying lists of polynomials. I don't know how useful that is, but that looks like what it's doing.