  
  [1X5 Ring Maps[0X
  
  A  [5Xhomalg[0m  ring  map is a data structure for maps between finitely generated
  rings.  [5Xhomalg[0m 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
  [5XSheaves[0m provides methods for ring maps of (finitely generated) affine rings.
  
  
  [1X5.1 Ring Maps: Category and Representations[0X
  
  [1X5.1-1 IsHomalgRingMap[0m
  
  [2X> IsHomalgRingMap( [0X[3Xphi[0X[2X ) ___________________________________________[0XCategory
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m category of ring maps.
  
  [1X5.1-2 IsHomalgRingSelfMap[0m
  
  [2X> IsHomalgRingSelfMap( [0X[3Xphi[0X[2X ) _______________________________________[0XCategory
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m category of ring self-maps.
  
  (It is a subcategory of the [5XGAP[0m category [10XIsHomalgRingMap[0m.)
  
  [1X5.1-3 IsHomalgRingMapRep[0m
  
  [2X> IsHomalgRingMapRep( [0X[3Xphi[0X[2X ) __________________________________[0XRepresentation
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  The [5XGAP[0m representation of finitley presented [5Xhomalg[0m modules.
  
  (It is a representation of the [5XGAP[0m category [2XIsHomalgRingMap[0m ([14X5.1-1[0m).)
  
  
  [1X5.2 Ring Maps: Constructors[0X
  
  [1X5.2-1 RingMap[0m
  
  [2X> RingMap( [0X[3Ximages, S, T[0X[2X ) _________________________________________[0Xoperation
  [6XReturns:[0X  a [5Xhomalg[0m ring map
  
  This  constructor  returns  a  ring map (homomorphism) of finitely generated
  rings/algebras.  It  is  represented  by  the  images  [3Ximages[0m  of the set of
  generators  of  the  source  [5Xhomalg[0m ring [3XS[0m in terms of the generators of the
  target  ring  [3XT[0m  (--> [14X4.2[0m). Unless the source ring is free [13Xand[0m 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  [2XIsMorphism[0m  ([14X5.3-1[0m).  If  source and target are identical
  objects, and only then, the ring map is created as a selfmap.
  
  
  [1X5.3 Ring Maps: Properties[0X
  
  [1X5.3-1 IsMorphism[0m
  
  [2X> IsMorphism( [0X[3Xphi[0X[2X ) ________________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check  if  [3Xphi[0m  is  a  well-defined  map,  i.e.  independent of all involved
  presentations.
  
  [1X5.3-2 IsIdentityMorphism[0m
  
  [2X> IsIdentityMorphism( [0X[3Xphi[0X[2X ) ________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m ring map [3Xphi[0m is the identity morphism.
  
  [1X5.3-3 IsMonomorphism[0m
  
  [2X> IsMonomorphism( [0X[3Xphi[0X[2X ) ____________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m ring map [3Xphi[0m is a monomorphism.
  
  [1X5.3-4 IsEpimorphism[0m
  
  [2X> IsEpimorphism( [0X[3Xphi[0X[2X ) _____________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m ring map [3Xphi[0m is an epimorphism.
  
  [1X5.3-5 IsIsomorphism[0m
  
  [2X> IsIsomorphism( [0X[3Xphi[0X[2X ) _____________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m ring map [3Xphi[0m is an isomorphism.
  
  [1X5.3-6 IsAutomorphism[0m
  
  [2X> IsAutomorphism( [0X[3Xphi[0X[2X ) ____________________________________________[0Xproperty
  [6XReturns:[0X  [10Xtrue[0m or [10Xfalse[0m
  
  Check if the [5Xhomalg[0m ring map [3Xphi[0m is an automorphism.
  
  
  [1X5.4 Ring Maps: Attributes[0X
  
  [1X5.4-1 Source[0m
  
  [2X> Source( [0X[3Xphi[0X[2X ) ___________________________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m ring
  
  The source of the [5Xhomalg[0m ring map [3Xphi[0m.
  
  [1X5.4-2 Range[0m
  
  [2X> Range( [0X[3Xphi[0X[2X ) ____________________________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m ring
  
  The target (range) of the [5Xhomalg[0m ring map [3Xphi[0m.
  
  [1X5.4-3 DegreeOfMorphism[0m
  
  [2X> DegreeOfMorphism( [0X[3Xphi[0X[2X ) _________________________________________[0Xattribute
  [6XReturns:[0X  an integer
  
  The degree of the morphism [3Xphi[0m of graded rings.
  (no method installed)
  
  [1X5.4-4 CoordinateRingOfGraph[0m
  
  [2X> CoordinateRingOfGraph( [0X[3Xphi[0X[2X ) ____________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m ring
  
  The coordinate ring of the graph of the ring map [3Xphi[0m.
  
  [1X5.4-5 KernelSubmodule[0m
  
  [2X> KernelSubmodule( [0X[3Xphi[0X[2X ) __________________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m submodule
  
  The kernel ideal of the ring map [3Xphi[0m (as a submodule).
  
  [1X5.4-6 KernelEmb[0m
  
  [2X> KernelEmb( [0X[3Xphi[0X[2X ) ________________________________________________[0Xattribute
  [6XReturns:[0X  a [5Xhomalg[0m map
  
  The  embedding  of  the  kernel ideal [10XKernel[0m([3Xphi[0m) into the [10XSource[0m([3Xphi[0m), both
  viewed as modules over the ring R :=[10XSource[0m([3Xphi[0m) (cf. [2XKernel[0m ([14X5.5-1[0m)).
  
  
  [1X5.5 Ring Maps: Operations and Functions[0X
  
  [1X5.5-1 Kernel[0m
  
  [2X> Kernel( [0X[3Xphi[0X[2X ) ______________________________________________________[0Xmethod
  [6XReturns:[0X  a [5Xhomalg[0m module
  
  The kernel ideal of the ring map [3Xphi[0m as an abstract module.
  
