The IsPrime, IsPrimary, and IsMaximal commands test whether an ideal is prime, primary, or maximal, respectively.  An ideal is prime if $fg$ in J implies either f in J or g in J.  It is primary if $fg$ in J implies that some power of f or of g is in J, and it is maximal if it is not contained within a larger ideal, other than the entire polynomial ring.

Prime ideals are primary and radical. Maximal ideals are zero-dimensional and prime.  By convention, prime, primary, and maximal ideals must also be proper, meaning that they are not the entire polynomial ring.  The IsProper command can be used to test this condition separately.  An optional second argument allows you to override the ring variables.

Primality tests are performed over the domain implied by the coefficients - usually the rationals or the integers mod p.  Additional field extensions can be specified with an optional second argument k, which may be a single RootOf or radical, or a list or set of RootOfs and radicals.

Of particular interest is the fact that any ideal can be decomposed into the finite intersection of primary ideals. The PrimaryDecomposition command can be used to do this.  The solution set of a primary ideal is an irreducible affine variety.

The algorithms employed by these commands require polynomials over a perfect field.  Infinite fields of positive characteristic are not supported, and over finite fields only zero-dimensional ideals can be handled because the dimension reducing process generates infinite fields.

Note: In contrast with Groebner[IsProper], PolynomialIdeals[IsProper] does not consider the zero ideal $⟨0⟩$ to be proper.

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

$J≔⟨x^2+1\,y^2-2\,z^2+2⟩$

$J≔⟨x^2+1\,y^2-2\,z^2+2⟩$

$f≔z-xy$

$f≔-xy+z$

$g≔z+xy$

$g≔xy+z$

$f\mathbf{in}J\mathbf{or}g\mathbf{in}J$

$\mathrm{false}$

$fg\mathbf{in}J$

$\mathrm{true}$

$\mathrm{IsPrime}\left(J\right)$

$\mathrm{false}$

$\mathrm{PrimaryDecomposition}\left(J\right)$

$⟨x^2+1\,y^2-2\,z^2+2\,-xy+z⟩,⟨x^2+1\,y^2-2\,z^2+2\,xy+z⟩$

$K≔⟨{\left(x-yz\right)}^2\,y^2+z^3+1⟩$

$K≔⟨{\left(-yz+x\right)}^2\,z^3+y^2+1⟩$

$x-yz\mathbf{in}K$

$\mathrm{false}$

$\mathrm{IsPrime}\left(K\right)$

$\mathrm{false}$

$\mathrm{IsPrimary}\left(K\right)$

$\mathrm{true}$

$L≔⟨x^4-2\,y^2+z⟩$

$L≔⟨x^4-2\,y^2+z⟩$

$\mathrm{IsPrime}\left(L\right)$

$\mathrm{true}$

$\mathrm{IsMaximal}\left(L\right)$

$\mathrm{false}$

$\mathrm{HilbertDimension}\left(L\right)$

$1$

$\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left(z^2-2\right)\right)$

$\mathrm{\alpha }$

$\mathrm{IsPrime}\left(L\,\mathrm{\alpha }\right)$

$\mathrm{false}$

$\mathrm{PrimaryDecomposition}\left(L\,\left\{\mathrm{\alpha }\right\}\right)$

$⟨x^2-\mathrm{\alpha }\,y^2+z⟩,⟨x^2+\mathrm{\alpha }\,y^2+z⟩$

$J≔⟨x\,x+1⟩$

$J≔⟨x\,x+1⟩$

$\mathrm{IsProper}\left(J\right)$

$\mathrm{false}$

$\mathrm{Simplify}\left(J\right)$

$⟨1⟩$

Gianni, P.; Trager, B.; and Zacharias, G. "Grobner bases and primary decompositions of polynomial ideals." J. Symbolic Comput., Vol. 6 (1988): 149-167.