The Add, Multiply, and Quotient commands compute ideal sums, products, and quotients respectively.

Let $J=<\mathrm{f1},...,\mathrm{fs}>$ and $K=<\mathrm{g1},...,\mathrm{gt}>$ be two polynomial ideals. The ideal sum $J+K$ is the ideal $<\mathrm{f1},...,\mathrm{fs},\mathrm{g1},...,\mathrm{gt}>$.  The ideal product $JK$ is the ideal $<f_i{g}_j:1<=i<=s,1<=j<=t>$.  The ideal quotient $J\&colon;K$ is the set of all polynomials $f$ such that $fg\in J$ for all $g$ in $K$.

Add and Multiply accept any number of arguments. The set of variables is extended to include the variables of each ideal.  If the ideals cannot be put into a common polynomial ring, then an error is produced.  Add and Multiply do not make any effort to simplify their results. The Simplify command can be used for this purpose.

The Quotient command accepts exactly two arguments.  If both arguments are polynomial ideals, then the set of variables is extended to include the variables of both ideals.  If one or more arguments are polynomials $f$, then the Quotient command takes that to mean $⟨f⟩$ in an appropriate polynomial ring.

$\mathrm{with}\left(\mathrm{PolynomialIdeals}\right)\&colon;$

$J≔⟨x^3-1\&comma;y^2-3⟩$

$J≔⟨x^3-1\&comma;y^2-3⟩$

$K≔⟨x^2-z⟩$

$K≔⟨x^2-z⟩$

$S≔\mathrm{Add}\left(J\&comma;K\right)$

$S≔⟨x^3-1\&comma;y^2-3\&comma;x^2-z⟩$

$\mathrm{IdealInfo}:-\mathrm{Variables}\left(S\right)$

$\left\{x\&comma;y\&comma;z\right\}$

$P≔\mathrm{Multiply}\left(J\&comma;K\right)$

$P≔⟨\left(x^3-1\right)\left(x^2-z\right)\&comma;\left(y^2-3\right)\left(x^2-z\right)⟩$

$\mathrm{IdealInfo}:-\mathrm{Variables}\left(P\right)$

$\left\{x\&comma;y\&comma;z\right\}$

$\mathrm{Quotient}\left(J\&comma;⟨x^2+x+1⟩\right)$

$⟨x-1\&comma;y^2-3⟩$

$\mathrm{Quotient}\left(J\&comma;x^2+x+1\right)$

$⟨x-1\&comma;y^2-3⟩$

$Q≔\mathrm{Quotient}\left(x^2-1\&comma;⟨x-1\&comma;\mathrm{characteristic}=2⟩\right)$

$Q≔⟨x+1⟩$

$\mathrm{IdealInfo}:-\mathrm{Characteristic}\left(Q\right)$

$2$