Elements of Modules

Creation of Elements

M ! v: ModDed, SeqEnum -> ModDedElt

M ! v: ModDed, ModDedElt -> ModDedElt

M ! v: ModDed, ModElt -> ModDedElt

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 a Dedekind domain which is compatible with \(M\).

Example: Coercion into Quotient Modules (ex-2c7014)

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Let Mod and its submodules and quotient modules be as in the sub and quotient module example above.

> m := 4*Mod.1;
> m;
(4/1*M.1 0)
> Q1!m;
( 4/1*M.1 0 )
> Q2!m;
( 4/1*M.1 0 )
> m := Mod!m;
> Q3!m;
( 4/1*M.1 0 )
> Q4!m;
( 4/1*M.1 0 )
> S1!m;
>> S1!m;
     ^
Runtime error in `!': Illegal coercion
LHS: ModDed
RHS: ModDedElt
> S1!Mod!V!0;
(  )
> S2!Mod!Mod.2;
( M.1 )
> S3!Mod!(4*Mod.1);
( 4/1*M.1 0 )

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Arithmetic with Elements

Basic arithmetic can be performed with elements of a module over a Dedekind domain.

x + y: ModDedElt, ModDedElt -> ModDedElt

The sum of the module elements.

x - y: ModDedElt, ModDedElt -> ModDedElt

The difference of the module elements.

u * c: ModDedElt, RngElt -> ModDedElt

c * u: RngElt, ModDedElt -> ModDedElt

The product of the module element \(u\) and the ring element \(c\).

u / c: ModDedElt, RngElt -> ModDedElt

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

I * u: RngOrdIdl, ModDedElt -> ModDed

I * u: RngFunOrdIdl, ModDedElt -> ModDed

u * I: ModDedElt, RngOrdIdl -> ModDed

u * I: ModDedElt, RngFunOrdIdl -> ModDed

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

Other Functions on Elements

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

x eq y: ModDedElt, ModDedElt -> Bool

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

IsZero(a): ModDedElt -> BoolElt

Returns whether the module element \(a\) is zero.

ElementToSequence(a): ModDedElt -> SeqEnum

Eltseq(a): ModDedElt -> SeqEnum

The module element \(a\) expressed as a sequence.