Module Elements

Construction

elt< M | a₁, ..., aₙ >: ModRng, List -> ModRngElt

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

M ! Q: ModRng, [RngElt] -> ModRngElt

Given the module \(M\) with underlying vector space \(K^n\), and a sequence \(Q = [a_1, \ldots, a_n]\) with universe \(K\), construct the element \(m = (a_1, \ldots, a_n)\) of \(M\). Note that if \(m\) is not an element of \(M\), an error will result.

Zero(M): ModRng -> 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): ModRngElt -> [RngElt]

Eltseq(u): ModRngElt -> [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: ModRngElt, AlgElt -> ModRngElt

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: ModRngElt, ModRngElt -> ModRngElt

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

- u: ModRngElt -> ModRngElt

Additive inverse of the element \(u\).

u - v: ModRngElt, ModRngElt -> ModRngElt

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

k * u: RngElt, ModRngElt -> ModRngElt

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

u * k: ModRngElt, RngElt -> ModRngElt

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

u / k: ModRngElt, RngElt -> ModRngElt

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

Indexing

u[i]: ModRngElt, RngIntElt -> RngElt

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

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

Given an element \(u\) belonging to a submodule \(M\) of the \(R\)-module \(T = R^{(n)}\), a positive integer \(i\), \(1 \leq i\leq n\), 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): ModRngElt -> BoolElt

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

Support(u): ModRngElt -> { RngIntElt }

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