MAGMA Computational Algebra System

Submodules and Quotient Modules

The following functions allow the construction of submodules and quotient modules and access to essential properties.

Subsections

• Creation
• Module Bases

Creation

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

Given a module M over a ring R, 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:

(a)

An element of M;

(b)

A set or sequence of elements of M;

(c)

A submodule of M;

(d)

A set or sequence of submodules of M.

A morphism is stored from the resulting submodule S into M, such that S.i is mapped to the i-th generator given in the above list.

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

Given a module M over a ring R, 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:

(a)

An element of M;

(b)

A set or sequence of elements of M;

(c)

A submodule of M;

(d)

A set or sequence of submodules of M.

A morphism is stored from M onto the resulting quotient module Q.

Submodule(I) : RngMPol -> ModMPol

Given an ideal I of a polynomial ring R, return the submodule of R¹ generated by I.

QuotientModule(I) : RngMPol -> ModMPol

Given an ideal I of a polynomial ring R, return the quotient module R¹/I.

GradedModule(I) : RngMPol -> ModMPol

Given a homogeneous ideal I of a ring R, return the graded quotient module R¹/I.

Module Bases

The following functions allow one to manipulate the bases of modules. Note that a Gröbner basis for a module will be automatically generated when necessary; the Groebner procedure just allows explicit immediate construction of the Gröbner basis.

Basis(M) : ModMPol -> RngMPolElt

Given a module M, return the current basis (whether it has been converted to a Gröbner basis or not) of M.

BasisElement(M, i) : ModMPol, RngIntElt -> RngMPolElt

Given a module M together with an integer i, return the i-th element of the current basis of M. Note that this is not the same as M.i.

BasisMatrix(M) : ModMPol -> ModMatRngElt

Given a module M, return the basis matrix of M, which is a k by r matrix over R, where k is the length of the basis of M and r is the degree of M.

Groebner(M) : ModMPol ->

(Procedure.) Explicitly force a Gröbner basis for the module M to be constructed.