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### 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).)

#### 9.3 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


#### 9.5 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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