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### 4 Ring Maps

A homalg ring map is a data structure for maps between finitely generated rings. homalg more or less provides the basic declarations and installs the generic methods for ring maps, but it is up to other high level packages to install methods applicable to specific rings. For example, the package Sheaves provides methods for ring maps of (finitely generated) affine rings.

#### 4.1 Ring Maps: Category and Representations

##### 4.1-1 IsHomalgRingMap
 ‣ IsHomalgRingMap( phi ) ( category )

Returns: true or false

The GAP category of ring maps.

##### 4.1-2 IsHomalgRingSelfMap
 ‣ IsHomalgRingSelfMap( phi ) ( category )

Returns: true or false

The GAP category of ring self-maps.

(It is a subcategory of the GAP category IsHomalgRingMap.)

##### 4.1-3 IsHomalgRingMapRep
 ‣ IsHomalgRingMapRep( phi ) ( representation )

Returns: true or false

The GAP representation of homalg ring maps.

(It is a representation of the GAP category IsHomalgRingMap (4.1-1).)

#### 4.2 Ring Maps: Constructors

##### 4.2-1 RingMap
 ‣ RingMap( images, S, T ) ( operation )

Returns: a homalg ring map

This constructor returns a ring map (homomorphism) of finitely generated rings/algebras. It is represented by the images images of the set of generators of the source homalg ring S in terms of the generators of the target ring T (--> 3.2). Unless the source ring is free and given on free ring/algebra generators the returned map will cautiously be indicated using parenthesis: "homomorphism". To verify if the result is indeed a well defined map use IsMorphism (4.3-1). If source and target are identical objects, and only then, the ring map is created as a selfmap.

#### 4.3 Ring Maps: Properties

##### 4.3-1 IsMorphism
 ‣ IsMorphism( phi ) ( property )

Returns: true or false

Check if phi is a well-defined map, i.e. independent of all involved presentations.

##### 4.3-2 IsIdentityMorphism
 ‣ IsIdentityMorphism( phi ) ( property )

Returns: true or false

Check if the homalg ring map phi is the identity morphism.

##### 4.3-3 IsMonomorphism
 ‣ IsMonomorphism( phi ) ( property )

Returns: true or false

Check if the homalg ring map phi is a monomorphism.

##### 4.3-4 IsEpimorphism
 ‣ IsEpimorphism( phi ) ( property )

Returns: true or false

Check if the homalg ring map phi is an epimorphism.

##### 4.3-5 IsIsomorphism
 ‣ IsIsomorphism( phi ) ( property )

Returns: true or false

Check if the homalg ring map phi is an isomorphism.

##### 4.3-6 IsAutomorphism
 ‣ IsAutomorphism( phi ) ( property )

Returns: true or false

Check if the homalg ring map phi is an automorphism.

#### 4.4 Ring Maps: Attributes

##### 4.4-1 Source
 ‣ Source( phi ) ( attribute )

Returns: a homalg ring

The source of the homalg ring map phi.

##### 4.4-2 Range
 ‣ Range( phi ) ( attribute )

Returns: a homalg ring

The target (range) of the homalg ring map phi.

##### 4.4-3 DegreeOfMorphism
 ‣ DegreeOfMorphism( phi ) ( attribute )

Returns: an integer

The degree of the morphism phi of graded rings.
(no method installed)

##### 4.4-4 CoordinateRingOfGraph
 ‣ CoordinateRingOfGraph( phi ) ( attribute )

Returns: a homalg ring

The coordinate ring of the graph of the ring map phi.

#### 4.5 Ring Maps: Operations and Functions

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