Arithmetic with Modules

Some arithmetic operations can be carried out involving modules and their elements and compatible ideals to gain more modules.

I * M: RngOrdIdl, ModDed -> ModDed

I * M: RngFunOrdIdl, ModDed -> ModDed

M * I: ModDed, RngOrdIdl -> ModDed

M * I: ModDed, RngFunOrdIdl -> ModDed

The module generated by the products of the ideals of the module \(M\) with \(I\).

M1 + M2: ModDed, ModDed -> ModDed

The union of the modules \(M1\) and \(M2\).

DirectSum(M1, M2): ModDed, ModDed -> ModDed, Map, Map, Map, Map

DirectSum(S): [ModDed] -> ModDed, [Map], [Map]

The direct sum \(D\) of the modules \(M1\) and \(M2\) or the modules in the sequence \(S\) together with the embedding maps \(M1 \rightarrow D, M2 \rightarrow D, \ldots\) and the projection maps \(D \rightarrow M1, D \rightarrow M2, \ldots\) returned in sequences only when a sequence \(S\) is input.

u * I: ModDedElt, RngOrdIdl -> ModDed

u * I: ModDedElt, RngFunOrdIdl -> ModDed

I * u: RngOrdIdl, ModDedElt -> ModDed

I * u: RngFunOrdIdl, ModDedElt -> ModDed

The module containing elements which are products of the Dedekind module element \(u\) and an element lying in the ideal \(I\).

Example: Module Arithmetic and Predicates (ex-79759b)

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Some module predicates and arithmetic are shown below.

> P<x> := PolynomialRing(Integers());
> K :=  NumberField([x^5 + 3, x^2 + 2]);
> M := MaximalOrder(K);
> Vs := RModule(M, 2);
> s := [Vs | [1, 3], [2, 3]];
> Mods := Module(s);
> sMods := sub<Mods | Mods!Vs![1, 3]>;
> Mods eq sMods;
false
> [1, 0] in Mods;
true
> [1, 0] in sMods;
false
> sMods subset Mods;
true
> Vs := RSpace(M, 2);
> s := [Vs | [Random(M, 3), 3], [2, Random(M, 2)]];
> Mods := Module(s);
> sMods := sub<Mods | Mods!s[1]>;
> (7*M + 11*K.1*M)*sMods;
Module over Maximal Equation Order with defining polynomial x^5 + [3, 0] over
its ground order
 generated by:
Ideal of M
Two element generators:
    7/1*$.1*M.1
    11/1*$.1*M.2 * ( M.1 + (-$.1 - 2/1*$.2)*M.2 + $.2*M.3 + (2/1*$.1 +
2/1*$.2)*M.5 3/1*$.1*M.1 )
 in echelon form:
Ideal of M
Two element generators:
    21/1*$.1*M.1
    33/1*$.1*M.2 * ( 1/3*$.1*M.1 + (-1/3*$.1 - 2/3*$.2)*M.2 + 1/3*$.2*M.3 +
(2/3*$.1 + 2/3*$.2)*M.5 M.1 )
> Mods + sMods;
Module over Maximal Equation Order with defining polynomial x^5 + [3, 0] over
its ground order
 generated by: (in echelon form)
Ideal of M
Two element generators:
    148919257164/1*$.1*M.1
    (148919257048/1*$.1 + 26/1*$.2)*M.1 + (32/1*$.1 + 148919257015/1*$.2)*M.2 +
    (196/1*$.1 + 148919257117/1*$.2)*M.3 + (148919257163/1*$.1 + 76/1*$.2)*M.4 +
    (148919257077/1*$.1 + 148919257116/1*$.2)*M.5 * ( M.1 0 )
Ideal of M
Two element generators:
    3/1*$.1*M.1
    ($.1 - $.2)*M.1 + -M.3 + M.4 + ($.1 + $.2)*M.5 * ( (-19866460521/1*$.1 -
33/1*$.2)*M.1 + (1/3*$.1 + 1/3*$.2)*M.4 + (1/3*$.1 + 1/3*$.2)*M.5 M.1 )
> 4*sMods;
Module over Maximal Equation Order with defining polynomial x^5 + [3, 0] over
its ground order
 generated by:
Principal Ideal of M
Generator:
    4/1*$.1*M.1 * ( (2/1*$.1 - 2/1*$.2)*M.1 + ($.1 + 2/1*$.2)*M.2 + (2/1*$.1 +
2/1*$.2)*M.3 + (2/1*$.1 - $.2)*M.5 3/1*$.1*M.1 )
 in echelon form:
Principal Ideal of M
Generator:
    12/1*$.1*M.1 * ( (2/3*$.1 - 2/3*$.2)*M.1 + (1/3*$.1 + 2/3*$.2)*M.2 +
(2/3*$.1 + 2/3*$.2)*M.3 + (2/3*$.1 - 1/3*$.2)*M.5 M.1 )

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