If we get a zero remainder, then the numerator should belong to
the ideal generated by the denominators::
- sage: set_random_seed()
sage: R = PolynomialRing(QQ, 'x,y,z')
sage: x,y,z = R.gens()
sage: s = ZZ.random_element(1,5).abs()
times the denominators, and the remainder's monomials aren't divisible
by the leading term of any denominator::
- sage: set_random_seed()
sage: R = PolynomialRing(QQ, 'x,y,z')
sage: s = ZZ.random_element(1,5).abs()
sage: gs = [ R.random_element() for idx in range(s) ]
should always get a zero remainder if we divide an element of a
monomial ideal by its generators::
- sage: set_random_seed()
sage: R = PolynomialRing(QQ,'x,y,z')
sage: gs = R.random_element().monomials()
sage: I = R.ideal(gs)