The IdealMembership command tests elements for membership in an ideal.  If the first argument is a list or set of polynomials, IdealMembership returns true if and only if all of the elements f are members of the ideal.  The criterion for ideal membership is Groebner[NormalForm](f, J) = 0.

The IdealContainment command tests whether ideals are contained within one another, and can test sequences of containments from left to right.  It returns true if and only if all containments are valid. For example, IdealContainment(J, K, L) tests whether J is contained in K and K is contained in L.

A particularly useful form of the IdealContainment command is IdealContainment(J, K, J), which tests whether the ideals J and K are equal.

The functionality of these commands is also available through the in and subset operators, see PolynomialIdeals[Operators] for more information.

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\:$

$J≔⟨x^2-y\,y^2-x⟩$

$J≔⟨y^2-x\,x^2-y⟩$

$K≔⟨x^4-y^2\,y^3-x^3⟩$

$K≔⟨-x^3+y^3\,x^4-y^2⟩$

$f≔\mathrm{expand}\left(\left(x^2-1\right)\left(x^2-y\right)-\left(xy-1\right)\left(y^2+x\right)\right)$

$f≔x^4-x{y}^3-2x^{2}y-x^2+y^2+x+y$

$\mathrm{IdealMembership}\left(f\,J\right)$

$\mathrm{true}$

$f\mathbf{in}J$

$\mathrm{true}$

$\mathrm{IdealMembership}\left(f\,K\right)$

$\mathrm{false}$

$\mathrm{IdealContainment}\left(K\,J\right)$

$\mathrm{true}$

$K\mathbf{subset}J$

$\mathrm{true}$

$L≔⟨-y^2+y^8\,x{y}^2-y^7\,-y^3+x^3⟩$

$L≔⟨y^8-y^2\,x^3-y^3\,-y^7+x{y}^2⟩$

$\mathrm{IdealContainment}\left(K\,L\,K\right)$

$\mathrm{true}$