Elements of Modules

Subsections

• Creation of Elements
• Arithmetic with Elements
• Other Functions on Elements

Creation of Elements

M ! v : ModOrd, SeqEnum -> ModOrdElt

M ! v : ModOrd, ModOrdElt -> ModOrdElt

M ! v : ModOrd, ModElt -> ModOrdElt

Coerce v into an element of M. v can be a sequence of length dimension of M, a module element or vector or an element of another module over an order which is compatible with M.

Example ModOrd_coerce-quo (H65E4)

> m := 4*Mod.1;
> m;
V: ([4, 0] [0, 0] [0, 0])
> Q1!m;
([0, 0] [0, 0] [0, 0])
> Q2!m;
([4, 0] [0, 0] [0, 0])
> m := Mod!m;
> Q3!m;
([4, 0] [0, 0] [0, 0])
> Q4!m;
([4, 0] [0, 0] [0, 0])

> S1!m;
>> S1!m;
     ^
Runtime error in '!': Illegal coercion
> S1!Mod!V!0;
([0, 0] [0, 0] [0, 0])
> S2!Mod!Mod.2;
([0, 0] [1, 0] [0, 0])
> S3!Mod!(4*Mod.1);
([4, 0] [0, 0] [0, 0])

Arithmetic with Elements

Basic arithmetic can be performed with elements of a module over an order.

x + y : ModOrdElt, ModOrdElt -> ModOrdElt

The sum of the module elements.

x - y : ModOrdElt, ModOrdElt -> ModOrdElt

The difference of the module elements.

u * c : ModOrdElt, RngElt -> ModOrdElt

c * u : RngElt, ModOrdElt -> ModOrdElt

The product of the module element u and the ring element c.

u / c : ModOrdElt, RngElt -> ModOrdElt

The product of u and 1/c if it lies in the parent module of u.

I * u : RngOrdIdl, ModOrdElt -> ModOrd

u * I : ModOrdElt, RngOrdIdl -> ModOrd

The module containing elements which are products of u and an element lying in I.

Other Functions on Elements

Elements of modules over order can be tested for equality and represented as a sequence.

x eq y : ModOrdElt, ModOrdElt -> Bool

Return true if x and y are the same element of a module.

ElementToSequence(a) : ModOrdElt -> SeqEnum

Eltseq(a) : ModOrdElt -> SeqEnum

The module element a expressed as a sequence.