Other Functions on Modules

Intersections of modules can taken. Several other functions are also available.

M1 meet M2: ModDed, ModDed -> ModDed

Return the intersection of the modules \(M1\) and \(M2\).

Dual(M): ModDed -> ModDed

The module dual to \(M\).

ElementaryDivisors(M, N): ModDed, ModDed -> SeqEnum

The elementary divisors (ideals) of the torsion part of the quotient \(R\)-module \(M/N\): For \(N\subseteq M\) we get

\[T(M/N) \cong \oplus_{i=1}^n R/{{\cal A}_i}\]

The \({{\cal A}_i}\) are unique if we require \(R\subseteq {{\cal A}_1} \subseteq \cdots \subseteq {{\cal A}_n}\). The \({{\cal A}_i}\) are called the elementary divisors (or elementary ideals) of \(M/N\). This corresponds to the Smith normal form for integral matrices.

SteinitzClass(M): ModDed -> RngOrdIdl

The Steinitz class of the module \(M\).

SteinitzForm(M): ModDed -> ModDed

The Steinitz (almost–free) form of the module \(M\).

Example: Basis Other (ex-69678f)

Some bases and other functions are demonstrated below.

> P<x> := PolynomialRing(Rationals());
> P<y> := PolynomialRing(P);
> F<c> := FunctionField(y^3 - x^3*y^2 + y - x^7);
> M := MaximalOrderFinite(F);
> Vs := RSpace(M, 2);
> s := [Vs | [1, Random(M, 3)], [Random(M, 3), 3]];
> Mods := Module(s);
> qMods := quo<Mods | Mods!s[2]>;
> Basis(Mods);
[
    ([ 1, 0, 0 ] [ 2*x^2 - 3*x + 1/2, -2/3*x^2 - x + 1/3, 2/3*x^2 - 2*x + 1/2
        ]),
    ([ -x^2 + x + 2/3, -1/3*x^2 - 1, -3/2*x^2 - 2/3*x + 1 ] [ 3, 0, 0 ])
]
> Basis(qMods);
[
    ([ 1, 0, 0 ] [ 2*x^2 - 3*x + 1/2, -2/3*x^2 - x + 1/3, 2/3*x^2 - 2*x + 1/2
        ]),
    ([ -x^2 + x + 2/3, -1/3*x^2 - 1, -3/2*x^2 - 2/3*x + 1 ] [ 3, 0, 0 ])
]
> PseudoBasis(Mods) eq PseudoBasis(qMods);
> Vs := RModule(M, 2);
> s := [Vs | [Random(M, 3), Random(M, 3)], [2, 3]];
> Mods := Module(s);
> sMods := sub<Mods | Mods!s[1]>;
> Mods meet sMods;
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Ideal of M
Generator:
(x + 1)*c^2 + (-3/2*x^2 - 3/2*x)*c + 1/2*x^2 + 2/3*x + 1/2
> ElementaryDivisors(Mods, sMods);
[ Ideal of M
Basis:
[1 0 0]
[0 1 0]
[0 0 1], Ideal of M
Generator:
0 ]
> Dual(Mods);
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Fractional ideal of M
Generator:
(-3*x^4 + 21*x^3 + 20*x^2 + 80*x - 387)/(x^17 - 24*x^16 + 192*x^15 - 510*x^14 -
    94/3*x^13 + 304/3*x^12 + 902/3*x^11 - 3109/3*x^10 - 389/9*x^9 + 664/9*x^8 +
    1094/3*x^7 - 1540/3*x^6 - 101/9*x^5 + 128/3*x^4 + 2000/27*x^3 - 4361/9*x^2 -
    427*x + 2056)*c^2 + (-3*x^9 + 48*x^8 - 189*x^7 - 24*x^6 + 3*x^5 - 48*x^4 +
    195*x^3 - 3*x^2 - x + 24)/(x^17 - 24*x^16 + 192*x^15 - 510*x^14 - 94/3*x^13
    + 304/3*x^12 + 902/3*x^11 - 3109/3*x^10 - 389/9*x^9 + 664/9*x^8 + 1094/3*x^7
    - 1540/3*x^6 - 101/9*x^5 + 128/3*x^4 + 2000/27*x^3 - 4361/9*x^2 - 427*x +
    2056)*c + (3*x^12 - 48*x^11 + 192*x^10 + 3*x^9 - 20*x^8 - 56*x^7 + 192*x^6 +
    3*x^5 - 5*x^4 - 8*x^3 + 272/3*x^2 + 128*x - 771)/(x^17 - 24*x^16 + 192*x^15
    - 510*x^14 - 94/3*x^13 + 304/3*x^12 + 902/3*x^11 - 3109/3*x^10 - 389/9*x^9 +
    664/9*x^8 + 1094/3*x^7 - 1540/3*x^6 - 101/9*x^5 + 128/3*x^4 + 2000/27*x^3 -
    4361/9*x^2 - 427*x + 2056) car Ideal of M
Generator:
1/3
> Dual(sMods);
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Fractional ideal of M
Generator:
(3/2*x^6 + 3*x^5 - 3/4*x^4 - 4*x^3 - 25/12*x^2 - 5/6*x - 1/2)/(x^17 + 3*x^16 +
    21/4*x^15 + 27/4*x^14 + 1/24*x^13 - 247/24*x^12 - 173/24*x^11 + 61/24*x^10 +
    305/72*x^9 + 17/72*x^8 - 4*x^7 - 53/12*x^6 - 1/8*x^5 + 23/6*x^4 +
    377/108*x^3 + 23/18*x^2 + 1/3*x + 1/8)*c^2 + (-5/2*x^9 - 5*x^8 - 1/4*x^7 +
    9/2*x^6 + 13/4*x^5 + 5/4*x^4 - 3/4*x^3 - 7/4*x^2 - 3/4*x)/(x^17 + 3*x^16 +
    21/4*x^15 + 27/4*x^14 + 1/24*x^13 - 247/24*x^12 - 173/24*x^11 + 61/24*x^10 +
    305/72*x^9 + 17/72*x^8 - 4*x^7 - 53/12*x^6 - 1/8*x^5 + 23/6*x^4 +
    377/108*x^3 + 23/18*x^2 + 1/3*x + 1/8)*c + (x^12 + 2*x^11 - 1/2*x^10 -
    7/2*x^9 - 8/3*x^8 - 5/12*x^7 + 11/4*x^6 + 19/4*x^5 - 1/4*x^4 - 25/6*x^3 -
    67/36*x^2 - 1/3*x - 1/4)/(x^17 + 3*x^16 + 21/4*x^15 + 27/4*x^14 + 1/24*x^13
    - 247/24*x^12 - 173/24*x^11 + 61/24*x^10 + 305/72*x^9 + 17/72*x^8 - 4*x^7 -
    53/12*x^6 - 1/8*x^5 + 23/6*x^4 + 377/108*x^3 + 23/18*x^2 + 1/3*x + 1/8)
> SteinitzClass(Mods) eq SteinitzClass(sMods);
false
> SteinitzForm(Mods);
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Ideal of M
Generator:
3 car Ideal of M
Generator:
1
> SteinitzForm(sMods);
Module over Maximal Equation Order of F over Univariate Polynomial Ring in x
over Rational Field
Ideal of M
Generator:
1

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