Modules over Dedekind domains can be created from orders of number fields and function fields and combinations of ideals and vector space elements. Submodules and quotient modules by submodules can also be created.
Create the free module On where O is a Dedekind domain.
Create the relative order O as a module over its coefficient ring. Also returns the map from the resulting module into O.
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.
Create a module from the sequence of tuples of ideals of a Dedekind domain and ModElts with entries in the Dedekind domain 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 ModElt. Also returns the map from the vector space into the module.
Create the module which is equal to the direct sum of the ideals in the sequence.
Create the module which is freely generated by the elements of the sequence S. The elements of the sequence must be ModElts with entries in a Dedekind domain or field of fractions of a Dedekind domain. Also returns the map from the vector space into the module.
> x := ext<Integers()|>.1;
> M := MaximalOrder(x^2 + 5);
> Module(M, 5);
Module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
generated by: (in echelon form)
Principal Ideal of M
Generator:
M.1 * ( M.1 0 0 0 0 )
Principal Ideal of M
Generator:
M.1 * ( 0 M.1 0 0 0 )
Principal Ideal of M
Generator:
M.1 * ( 0 0 M.1 0 0 )
Principal Ideal of M
Generator:
M.1 * ( 0 0 0 M.1 0 )
Principal Ideal of M
Generator:
M.1 * ( 0 0 0 0 M.1 )
> I := 1/5*M;
> Module([I, I^3, I^8]);
Module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
generated by: (in echelon form)
Fractional Principal Ideal of M
Generator:
1/5*M.1 * ( M.1 0 0 )
Fractional Principal Ideal of M
Generator:
1/125*M.1 * ( 0 M.1 0 )
Fractional Principal Ideal of M
Generator:
1/390625*M.1 * ( 0 0 M.1 )
> V := RModule(M, 3);
> Module([<I, V![0, 1, 0]>, <I^4, V![2, 3, 5]>]);
Module over Maximal Equation Order with defining polynomial x^2 + 5 over Z
Fractional Principal Ideal of M
Generator:
1/5*M.1 car Fractional Principal Ideal of M
Generator:
1/125*M.1
The same can be done using orders of function fields.
> P<x> := PolynomialRing(Rationals()); > P<y> := PolynomialRing(P); > F<c> := FunctionField(x^2 - y); > M := MaximalOrderFinite(F); > Module(M, 5); Module over Maximal Equation Order of F over Univariate Polynomial Ring in x over Rational Field generated by: (in echelon form) Ideal of M Generator: 1 * ( 1 0 0 0 0 ) Ideal of M Generator: 1 * ( 0 1 0 0 0 ) Ideal of M Generator: 1 * ( 0 0 1 0 0 ) Ideal of M Generator: 1 * ( 0 0 0 1 0 ) Ideal of M Generator: 1 * ( 0 0 0 0 1 ) > I := 1/5*M; > Module([I, I^3, I^8]); Module over Maximal Equation Order of F over Univariate Polynomial Ring in x over Rational Field generated by: (in echelon form) Ideal of M Generator: 1/5 * ( 1 0 0 ) Ideal of M Generator: 1/125 * ( 0 1 0 ) Ideal of M Generator: 1/390625 * ( 0 0 1 ) > V := RModule(M, 3); > Module([<I, V![0, 1, 0]>, <I^4, V![2, 3, 5]>]); Integral Module over Maximal Equation Order of F over Univariate Polynomial Ring in x over Rational Field Ideal of M Generator: 1/5 car Ideal of M Generator: 1/125
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.
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.
> Mod := Module([V|[0,1,0], [4,4,0]]);
> S1 := sub<Mod | >;
> S1;
Integral Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
(0)
> Q1 := quo<Mod | Mod>;
> Q1;
Quotient of Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
by Integral Module over Maximal Equation Order with defining polynomial
x^2 + 5 over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
> S2 := sub<Mod | Mod.2>;
> S2;
Integral Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
M.1
> Q2 := quo<Mod | Mod.2>;
> Q2;
Quotient of Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
by Integral Module over Maximal Equation Order with defining polynomial
x^2 + 5 over Z
Principal Ideal of M
Generator:
M.1
> S3 := sub<Mod | 4*Mod.1, Mod.2>;
> S3;
Integral Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
> Q3 := quo<Mod | >;
> Q3;
Integral Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
> Q4 := quo<Mod | S1>;
> Q4;
Quotient of Module over Maximal Equation Order with defining polynomial x^2 + 5
over Z
Principal Ideal of M
Generator:
4/1*M.1 car Principal Ideal of M
Generator:
M.1
by Integral Module over Maximal Equation Order with defining polynomial
x^2 + 5 over Z
(0)