GAP (Modules) - Chapter 9: Module Elements

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9 Module Elements
9.1 Module Elements: Category and Representations
9.1-1 IsHomalgElement
9.1-2 IsElementOfAModuleGivenByAMorphismRep
9.2 Module Elements: Constructors
9.3 Module Elements: Properties
9.3-1 IsElementOfIntegers
9.4 Module Elements: Attributes
9.5 Module Elements: Operations and Functions
9.5-1 HomalgRing

### 9 Module Elements

An element of a module \(M\) is internally represented by a module map from the (distinguished) rank 1 free module to the module \(M\). In particular, the data structure for module elements automatically profits from the intrinsic realization of morphisms in the homalg project.

#### 9.1 Module Elements: Category and Representations

##### 9.1-1 IsHomalgElement

`‣ IsHomalgElement`( M ) | ( category )

Returns: `true` or `false`

The GAP category of module elements.

##### 9.1-2 IsElementOfAModuleGivenByAMorphismRep

`‣ IsElementOfAModuleGivenByAMorphismRep`( M ) | ( representation )

Returns: `true` or `false`

The GAP representation of elements of modules.

(It is a subrepresentation of `IsElementOfAnObjectGivenByAMorphismRep` (homalg: IsElementOfAnObjectGivenByAMorphismRep).)

#### 9.2 Module Elements: Constructors

#### 9.3 Module Elements: Properties

##### 9.3-1 IsElementOfIntegers

`‣ IsElementOfIntegers`( m ) | ( property )

Returns: `true` or `false`

Check if the m is an element of the integers viewed as a module over itself.

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> a := HomalgElement( HomalgMap( "[[2]]", 1 * ZZ, 1 * ZZ ) );
    2
    gap> IsElementOfIntegers( a );
    true
    gap> Z4 := ZZ / 4;
    Z/( 4 )
    gap> b := HomalgElement( HomalgMap( "[[-1]]", 1 * Z4, 1 * Z4 ) );
    |[ 3 ]|
    gap> IsElementOfIntegers( b );
    false

#### 9.4 Module Elements: Attributes

#### 9.5 Module Elements: Operations and Functions

##### 9.5-1 HomalgRing

`‣ HomalgRing`( m ) | ( operation )

Returns: a homalg ring

The homalg ring of the homalg module element m.

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> a := HomalgElement( HomalgMap( "[[2]]", 1 * ZZ, 1 * ZZ ) );
    2
    gap> IsIdenticalObj( ZZ, HomalgRing( a ) );
    true

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