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

Let  and  be two polynomial ideals. The ideal sum  is the ideal .  The ideal product  is the ideal .  The ideal quotient  is the set of all polynomials  such that  for all  in .

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 , then the Quotient command takes that to mean  in an appropriate polynomial ring.