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57 Modules

57.1 Generating modules

57.1-1 IsLeftOperatorAdditiveGroup

57.1-2 IsLeftModule

57.1-3 GeneratorsOfLeftOperatorAdditiveGroup

57.1-4 GeneratorsOfLeftModule

57.1-5 AsLeftModule

57.1-6 IsRightOperatorAdditiveGroup

57.1-7 IsRightModule

57.1-8 GeneratorsOfRightOperatorAdditiveGroup

57.1-9 GeneratorsOfRightModule

57.1-10 LeftModuleByGenerators

57.1-11 LeftActingDomain

57.1-1 IsLeftOperatorAdditiveGroup

57.1-2 IsLeftModule

57.1-3 GeneratorsOfLeftOperatorAdditiveGroup

57.1-4 GeneratorsOfLeftModule

57.1-5 AsLeftModule

57.1-6 IsRightOperatorAdditiveGroup

57.1-7 IsRightModule

57.1-8 GeneratorsOfRightOperatorAdditiveGroup

57.1-9 GeneratorsOfRightModule

57.1-10 LeftModuleByGenerators

57.1-11 LeftActingDomain

`‣ IsLeftOperatorAdditiveGroup` ( D ) | ( category ) |

A domain `D` lies in `IsLeftOperatorAdditiveGroup`

if it is an additive group that is closed under scalar multiplication from the left, and such that \(\lambda * ( x + y ) = \lambda * x + \lambda * y\) for all scalars \(\lambda\) and elements \(x, y \in D\) (here and below by scalars we mean elements of a domain acting on `D` from left or right as appropriate).

`‣ IsLeftModule` ( M ) | ( category ) |

A domain `M` lies in `IsLeftModule`

if it lies in `IsLeftOperatorAdditiveGroup`

, *and* the set of scalars forms a ring, *and* \((\lambda + \mu) * x = \lambda * x + \mu * x\) for scalars \(\lambda, \mu\) and \(x \in M\), *and* scalar multiplication satisfies \(\lambda * (\mu * x) = (\lambda * \mu) * x\) for scalars \(\lambda, \mu\) and \(x \in M\).

gap> V:= FullRowSpace( Rationals, 3 ); ( Rationals^3 ) gap> IsLeftModule( V ); true

`‣ GeneratorsOfLeftOperatorAdditiveGroup` ( D ) | ( attribute ) |

returns a list of elements of `D` that generates `D` as a left operator additive group.

`‣ GeneratorsOfLeftModule` ( M ) | ( attribute ) |

returns a list of elements of `M` that generate `M` as a left module.

gap> V:= FullRowSpace( Rationals, 3 );; gap> GeneratorsOfLeftModule( V ); [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ]

`‣ AsLeftModule` ( R, D ) | ( operation ) |

if the domain `D` forms an additive group and is closed under left multiplication by the elements of `R`, then `AsLeftModule( `

returns the domain `R`, `D` )`D` viewed as a left module.

gap> coll:= [[0*Z(2),0*Z(2)], [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)]]; [ [ 0*Z(2), 0*Z(2) ], [ Z(2)^0, 0*Z(2) ], [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] gap> AsLeftModule( GF(2), coll ); <vector space of dimension 2 over GF(2)>

`‣ IsRightOperatorAdditiveGroup` ( D ) | ( category ) |

A domain `D` lies in `IsRightOperatorAdditiveGroup`

if it is an additive group that is closed under scalar multiplication from the right, and such that \(( x + y ) * \lambda = x * \lambda + y * \lambda\) for all scalars \(\lambda\) and elements \(x, y \in D\).

`‣ IsRightModule` ( M ) | ( category ) |

A domain `M` lies in `IsRightModule`

if it lies in `IsRightOperatorAdditiveGroup`

, *and* the set of scalars forms a ring, *and* \(x * (\lambda + \mu) = x * \lambda + x * \mu\) for scalars \(\lambda, \mu\) and \(x \in M\), *and* scalar multiplication satisfies \((x * \mu) * \lambda = x * (\mu * \lambda)\) for scalars \(\lambda, \mu\) and \(x \in M\).

`‣ GeneratorsOfRightOperatorAdditiveGroup` ( D ) | ( attribute ) |

returns a list of elements of `D` that generates `D` as a right operator additive group.

`‣ GeneratorsOfRightModule` ( M ) | ( attribute ) |

returns a list of elements of `M` that generate `M` as a left module.

`‣ LeftModuleByGenerators` ( R, gens[, zero] ) | ( operation ) |

returns the left module over `R` generated by `gens`.

gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];; gap> V:= LeftModuleByGenerators( GF(16), coll ); <vector space over GF(2^4), with 3 generators>

`‣ LeftActingDomain` ( D ) | ( attribute ) |

Let `D` be an external left set, that is, `D` is closed under the action of a domain \(L\) by multiplication from the left. Then \(L\) can be accessed as value of `LeftActingDomain`

for `D`.

`‣ Submodule` ( M, gens[, "basis"] ) | ( function ) |

is the left module generated by the collection `gens`, with parent module `M`. If the string `"basis"`

is entered as the third argument then the submodule of `M` is created for which the list `gens` is known to be a list of basis vectors; in this case, it is *not* checked whether `gens` really is linearly independent and whether all in `gens` lie in `M`.

gap> coll:= [ [Z(2),0*Z(2)], [0*Z(2),Z(2)], [Z(2),Z(2)] ];; gap> V:= LeftModuleByGenerators( GF(16), coll );; gap> W:= Submodule( V, [ coll[1], coll[2] ] ); <vector space over GF(2^4), with 2 generators> gap> Parent( W ) = V; true

`‣ SubmoduleNC` ( M, gens[, "basis"] ) | ( function ) |

`SubmoduleNC`

does the same as `Submodule`

(57.2-1), except that it does not check whether all in `gens` lie in `M`.

`‣ ClosureLeftModule` ( M, m ) | ( operation ) |

is the left module generated by the left module generators of `M` and the element `m`.

gap> V:= LeftModuleByGenerators(Rationals, [ [ 1, 0, 0 ], [ 0, 1, 0 ] ]); <vector space over Rationals, with 2 generators> gap> ClosureLeftModule( V, [ 1, 1, 1 ] ); <vector space over Rationals, with 3 generators>

`‣ TrivialSubmodule` ( M ) | ( attribute ) |

returns the zero submodule of `M`.

gap> V:= LeftModuleByGenerators(Rationals, [[ 1, 0, 0 ], [ 0, 1, 0 ]]);; gap> TrivialSubmodule( V ); <vector space over Rationals, with 0 generators>

`‣ IsFreeLeftModule` ( M ) | ( category ) |

A left module is free as module if it is isomorphic to a direct sum of copies of its left acting domain.

Free left modules can have bases.

The characteristic (see `Characteristic`

(31.10-1)) of a free left module is defined as the characteristic of its left acting domain (see `LeftActingDomain`

(57.1-11)).

`‣ FreeLeftModule` ( R, gens[, zero][, "basis"] ) | ( function ) |

`FreeLeftModule( `

is the free left module over the ring `R`, `gens` )`R`, generated by the vectors in the collection `gens`.

If there are three arguments, a ring `R` and a collection `gens` and an element `zero`, then `FreeLeftModule( `

is the `R`, `gens`, `zero` )`R`-free left module generated by `gens`, with zero element `zero`.

If the last argument is the string `"basis"`

then the vectors in `gens` are known to form a basis of the free module.

It should be noted that the generators `gens` must be vectors, that is, they must support an addition and a scalar action of `R` via left multiplication. (See also Section 31.3 for the general meaning of "generators" in **GAP**.) In particular, `FreeLeftModule`

is *not* an equivalent of commands such as `FreeGroup`

(37.2-1) in the sense of a constructor of a free group on abstract generators. Such a construction seems to be unnecessary for vector spaces, for that one can use for example row spaces (see `FullRowSpace`

(61.9-4)) in the finite dimensional case and polynomial rings (see `PolynomialRing`

(66.15-1)) in the infinite dimensional case. Moreover, the definition of a "natural" addition for elements of a given magma (for example a permutation group) is possible via the construction of magma rings (see Chapter 65).

gap> V:= FreeLeftModule(Rationals, [[ 1, 0, 0 ], [ 0, 1, 0 ]], "basis"); <vector space of dimension 2 over Rationals>

`‣ Dimension` ( M ) | ( attribute ) |

A free left module has dimension \(n\) if it is isomorphic to a direct sum of \(n\) copies of its left acting domain.

(We do *not* mark `Dimension`

as invariant under isomorphisms since we want to call `UseIsomorphismRelation`

(31.13-3) also for free left modules over different left acting domains.)

gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> Dimension( V ); 2

`‣ IsFiniteDimensional` ( M ) | ( property ) |

is `true`

if `M` is a free left module that is finite dimensional over its left acting domain, and `false`

otherwise.

gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> IsFiniteDimensional( V ); true

`‣ UseBasis` ( V, gens ) | ( operation ) |

The vectors in the list `gens` are known to form a basis of the free left module `V`. `UseBasis`

stores information in `V` that can be derived form this fact, namely

`gens`are stored as left module generators if no such generators were bound (this is useful especially if`V`is an algebra),the dimension of

`V`is stored.

gap> V:= FreeLeftModule( Rationals, [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ] ] );; gap> UseBasis( V, [ [ 1, 0 ], [ 1, 1 ] ] ); gap> V; # now V knows its dimension <vector space of dimension 2 over Rationals>

`‣ IsRowModule` ( V ) | ( property ) |

A *row module* is a free left module whose elements are row vectors.

`‣ IsMatrixModule` ( V ) | ( property ) |

A *matrix module* is a free left module whose elements are matrices.

`‣ IsFullRowModule` ( M ) | ( property ) |

A *full row module* is a module \(R^n\), for a ring \(R\) and a nonnegative integer \(n\).

More precisely, a full row module is a free left module over a ring \(R\) such that the elements are row vectors of the same length \(n\) and with entries in \(R\) and such that the dimension is equal to \(n\).

Several functions delegate their tasks to full row modules, for example `Iterator`

(30.8-1) and `Enumerator`

(30.3-2).

`‣ FullRowModule` ( R, n ) | ( function ) |

is the row module

, for a ring `R`^`n``R` and a nonnegative integer `n`.

gap> V:= FullRowModule( Integers, 5 ); ( Integers^5 )

`‣ IsFullMatrixModule` ( M ) | ( property ) |

A *full matrix module* is a module \(R^{{[m,n]}}\), for a ring \(R\) and two nonnegative integers \(m\), \(n\).

More precisely, a full matrix module is a free left module over a ring \(R\) such that the elements are \(m\) by \(n\) matrices with entries in \(R\) and such that the dimension is equal to \(m n\).

`‣ FullMatrixModule` ( R, m, n ) | ( function ) |

is the matrix module

, for a ring `R`^[`m`,`n`]`R` and nonnegative integers `m` and `n`.

gap> FullMatrixModule( GaussianIntegers, 3, 6 ); ( GaussianIntegers^[ 3, 6 ] )

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