Element Construction and Operations

Subsections

• Construction of Module Elements
• Deconstruction of Module Elements
• Action of the Algebra on the Module
• Arithmetic with Module Elements
• Indexing
• Properties of Module Elements

Construction of Module Elements

elt< M | a_1, ..., a_n > : ModTupRng, List -> ModTupRngElt

Given a module M with underlying vector space K^((n)), and elements a_1, ..., a_n belonging to K, construct the element m = (a_1, ..., a_n) of M. Note that if m is not an element of M, an error will result.

M ! Q : ModTupRng, [RngElt] -> ModTupRngElt

Given the module M with underlying vector space K^n, and elements a_1, ..., a_n belonging to K, construct the element m = (a_1, ..., a_n) of M. Note that if m is not an element of M, an error will result.

Zero(M) : ModRng, RngIntElt -> ModRngElt

M ! 0 : ModRng, RngIntElt -> ModRngElt

The zero element for the A-module M.

Random(M) : ModRng -> ModRngElt

Given a module M defined over a finite ring or field, return a random vector.

Deconstruction of Module Elements

ElementToSequence(u) : ModTupRngElt -> [RngElt]

Eltseq(u) : ModTupRngElt -> [RngElt]

Given an element u belonging to the A-module M, return u in the form of a sequence Q of elements of K.

Action of the Algebra on the Module

u * a : ModTupElt, AlgElt -> ModTupElt

Given a vector u belonging to an A-module M, and an element a in A return the image of u under the action of a.

u * g : ModGrpElt, GrpElt -> ModGrpElt

Given a vector u belonging to an K[G]-module M, and an element g belonging to the group G, return the image of u under the action of K[G] on the module M.

Arithmetic with Module Elements

u + v : ModTupElt, ModTupElt -> ModTupElt

Sum of the elements u and v, where u and v lie in the same A-module M.

- u : ModTupElt -> ModTupElt

Additive inverse of the element u.

u - v : ModTupElt, ModTupElt -> ModTupElt

Difference of the elements u and v, where u and v lie in the same A-module M.

k * u : RngElt, ModTupElt -> ModTupElt

Given an A-module M, where A is a K-algebra, an element k in K, and an element u in M, return the scalar product x * k as an element of M.

u * k : ModTupElt, RngElt -> ModTupElt

Given an A-module M, where A is a K-algebra, an element k in K, and an element u in M, return the scalar product u * k as an element of M.

u / k : ModTupElt, RngElt -> ModTupElt

Given an A-module M, where A is a K-algebra, a non-zero element k in K, and an element u in M, return the scalar product u * (1/k) as an element of M.

Indexing

u[i] : ModTupRngElt, RngIntElt -> RngElt

Given an element u belonging to a submodule M of the R-module R^((n)) and a positive integer i, 1 <= i <= n, return the i-th component of u (as an element of the ring R).

u[i] := x : ModTupRngElt, RngIntElt, RngElt -> ModTupRngElt

Given an element u belonging to a submodule M of the R-module T = R^((n)), a positive integer i, 1 <= i <= n, and an element x of the ring R, redefine the i-th component of u to be x. The parent of u is changed to T (since the modified element u need not lie in M).

Properties of Module Elements

IsZero(u) : ModTupElt -> BoolElt

Returns true if the element u of the A-module M is the zero element.

Support(u) : ModTupRngElt -> { RngElt }

A set of integers giving the positions of the non-zero components of the vector u.