# 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: Ops Arith (ex-79759b)`

Some module predicates and arithmetic are shown below.

```magma
> 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 )

```