The following functions allow the construction of submodules and quotient modules. See above for the definition of submodules and quotient modules.

sub<M | L> : ModMPol, List -> ModMPol

Given a module M over a multivariate polynomial ring or quotient ring P, return the submodule of M (with the same quotient relations as M) generated by the elements of M specified by the list L. Each term of the list L must be an expression defining an object of one of the following types:

• An element of M;
• A set or sequence of elements of M;
• A submodule of M;
• A set or sequence of submodules of M.

Given a module M over a multivariate polynomial ring or quotient ring P, return the quotient module of M by the elements of M specified by the list L. Each term of the list L must be an expression defining an object of one of the following types:

• An element of M;
• A set or sequence of elements of M;
• A submodule of M;
• A set or sequence of submodules of M.

The resulting module Q has the quotient relations of M together with those specified in L. The terms of L must be compatible with M. Thus if M is free, the quotient relations for Q are obtained by all the terms of L which must all lie in a free module compatible with M.