Creation of Modules

Modules over orders can be created from orders and combinations of ideals and vector space elements. Submodules and quotient modules by submodules can also be created.

Module(O, n) : RngOrd, RngIntElt -> ModOrd

Create the free module O^n where O is a maximal order.

Module(O) : RngOrd -> ModOrd, Map

Create the relative order O as a module over its coefficient ring. Also returns the map from the resulting module into O.

Module(I) : RngOrdFracIdl -> ModOrd, Map

Create the ideal I of a relative order O as a module over the coefficient ring of the order. Also returns the map from the module into O.

Module(S) : SeqEnum[Tup] -> ModOrd, Map

Create a module from the sequence of tuples of ideals of an order and ModRngElts with entries in the order or its field of fractions. The elements of the resulting module will be the sum of products of an element of an ideal and the corresponding ModRngElt. Also returns the map from the vector space into the module.

Module(S) : SeqEnum[RngOrdFracIdl] -> ModOrd

Create the module which is equal to the direct sum of the ideals in the sequence.

Module(S) : SeqEnum[ModRngElt] -> ModOrd, Map, ModMatRngElt

Create the module generated by the elements of the sequence S. The elements of the sequence must be ModRngElts with entries in a maximal order or field of fractions of a maximal order. Also returns the map from the vector space into the module and in some cases the transformation matrix from the given information to the module in normal form as it is returned.

Example ModOrd_create (H65E1)

> x := ext<Integers()|>.1;
> M := MaximalOrder(x^2 + 5);
> Module(M, 5);
free module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
 of rank 5 and dimension 5

> I := 1/5*M;
> Module([I, I^3, I^8]);
module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
<[1, 0] / 5> car <[1, 0] / 125> car <[1, 0] / 390625>
> V := RModule(M, 3);
> Module([<I, V![0, 1, 0]>, <I^4, V![2, 3, 5]>]);
integral module over Maximal Equation Order with defining polynomial x^2 + 5 
over Z
 with (pseudo) basis in echelon form:
<[1, 0] / 5> * ( [0, 0] [1, 0] [0, 0] ) 
<[1, 0] / 125> * ( [2, 0] / 5 [3, 0] / 5 [1, 0] ) 
Mapping from: ModRng: V to integral module over Maximal Equation Order with 
defining polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[1, 0] / 5> * ( [0, 0] [1, 0] [0, 0] ) 
<[1, 0] / 125> * ( [2, 0] / 5 [3, 0] / 5 [1, 0] ) 

sub<M | m> : ModOrd, SeqEnum[ModOrdElt] -> ModOrd, Map

sub<M | m1, .., mn> : ModOrd, ModOrdElt, ..., ModOrdElt -> ModOrd, Map

Construct the submodule of the module M generated by the elements in the sequence or list of elements m. Also returns the inclusion map of the submodule into M.

quo<M | S> : ModOrd, ModOrd -> ModOrd, Map

quo<M | m> : ModOrd, SeqEnum[ModOrdElt] -> ModOrd, Map

quo<M | m1, .., mn> : ModOrd, ModOrdElt, ..., ModOrdElt -> ModOrd, Map

Construct the quotient of the module M by the submodule S or the submodule generated by the elements of the sequence or list of elements m. Also returns the inclusion map of the quotient module into M.

Example ModOrd_sub-quo (H65E2)

> Mod := Module([V|[0,1,0], [4,4,0]]);
> S1 := sub<Mod | >;
> S1;
integral free module over Maximal Equation Order with defining polynomial x^2 + 
    5 over Z
(0)

> Q1 := quo<Mod | Mod>;
> Q1;
Quotient of integral free module over Maximal Equation Order with defining 
polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] )  by integral free module over Maximal 
Equation Order with defining polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] ) 
> S2 := sub<Mod | Mod.2>;
> S2;
integral free module over Maximal Equation Order with defining polynomial x^2 + 
    5 over Z
 with (pseudo) basis in echelon form:
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] ) 
> Q2 := quo<Mod | Mod.2>;
> Q2;
Quotient of integral free module over Maximal Equation Order with defining 
polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] )  by integral free module over Maximal 
Equation Order with defining polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] ) 
> S3 := sub<Mod | 4*Mod.1, Mod.2>;
> S3;
integral free module over Maximal Equation Order with defining polynomial x^2 + 
    5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] ) 
> Q3 := quo<Mod | >;
> Q3;
integral free module over Maximal Equation Order with defining polynomial x^2 + 
    5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] ) 
> Q4 := quo<Mod | S1>;
> Q4;
Quotient of integral free module over Maximal Equation Order with defining 
polynomial x^2 + 5 over Z
 with (pseudo) basis in echelon form:
<[4, 0]> * ( [1, 0] [0, 0] [0, 0] ) 
<[1, 0]> * ( [0, 0] [1, 0] [0, 0] )  by integral free module over Maximal 
Equation Order with defining polynomial x^2 + 5 over Z
(0)