Elementary Functions

Various simple properties of a module can be retrieved using the following functions.

BaseRing(M) : ModDed -> Rng

CoefficientRing(M) : ModDed -> Rng

The dedekind domain which M is a module over.

Degree(M) : ModDed -> RngIntElt

The dimension of the vector space M embeds into.

Ngens(M) : ModDed -> RngIntElt

NumberOfGenerators(M) : ModDed -> RngIntElt

The minimum number of vectors and ideals which generate M.

M . i : ModDed, RngIntElt -> ModTupRngElt

The vector of the ith vector and ideal pair generating M.

Determinant(M) : ModDed -> RngOrdIdl

Determinant(M) : ModDed -> RngFunOrdIdl

The determinant of M.

Dimension(M) : ModDed -> RngIntElt

The dimension of the vector space spanned by M over its coefficient ring. This is the same as the number of generators of M.

Example ModDed_elementary (H58E3)

> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<c> := FunctionField(x^2 - y);
> M := MaximalOrderFinite(F);
> Vs := RModule(M, 2);
> s := [Vs | [1, 3], [2, 3]];
> Mods := Module(s);
> CoefficientRing(Mods);
Maximal Equation Order of F over Univariate Polynomial Ring in x over Rational
Field
> Mods.1;
(1 0)
> Determinant(Mods);
Ideal of M
Basis:
[1]
> Vs := RSpace(M, 2);
> s := [Vs | [1, 3], [2, 3]];
> Mods := Module(s);
> sMods := sub<Mods | Mods!Vs![1, 3]>;
> qMods := quo<Mods | sMods>;
> Degree(Mods);
2
> Ngens(Mods);
2
> Ngens(sMods);
1
> Degree(sMods);
2
> Degree(qMods);
2
> Ngens(qMods);
2
> Determinant(Mods);
Ideal of M
Basis:
[1]

> Determinant(sMods);
>>   Determinant(sMods);
               ^
Runtime error in 'Determinant': Module must be square
> Determinant(qMods);
Ideal of M
Basis:
[1]