Some arithmetic operations can be carried out involving modules and their elements and compatible ideals to gain more modules.
The module generated by the products of the ideals of the module M with I.
The union of the modules M1 and M2.
The direct sum D of the modules M1 and M2 or the modules in the sequence S together with the embedding maps M1 -> D, M2 -> D, ... and the projection maps D -> M1, D -> M2, ... returned in sequences only when a sequence S is input.
The module containing elements which are products of the Dedekind module element u and an element lying in the ideal I.
> 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 )