Some more tweaks.

src/Modules/ModuleTypes.jl

@@ -545,13 +545,13 @@ When computed, the corresponding matrix (via `matrix()`) and inverse isomorphism
     @assert all(x->parent(x) === G, a)
     @assert length(a) == ngens(F)
     r = new{typeof(F), typeof(G), Nothing}()
+    a=Vector{elem_type(G)}(a)
     function im_func(x::AbstractFreeModElem)
      #if r.generators_map_to_generators === nothing
      #  r.generators_map_to_generators = images_of_generators(r) == gens(codomain(r))
      #end
       r.generators_map_to_generators === true && return codomain(r)(coordinates(x))
-      v = images_of_generators(r)
-      return sum(a*v[i] for (i, a) in coordinates(x); init=zero(codomain(r)))
+      return sum(b*a[i] for (i, b) in coordinates(x); init=zero(codomain(r)))
     end
     function pr_func(x)
       @assert parent(x) === G
@@ -572,14 +572,14 @@ When computed, the corresponding matrix (via `matrix()`) and inverse isomorphism
     @assert length(a) == ngens(F)
     @assert h(one(base_ring(F))) == one(base_ring(G))
     r = new{typeof(F), T2, RingMapType}()
+    a=Vector{elem_type(G)}(a)
     function im_func(x::AbstractFreeModElem)
       iszero(x) && return zero(codomain(r))
      #if r.generators_map_to_generators === nothing
      #  r.generators_map_to_generators = images_of_generators(r) == gens(codomain(r))
      #end
       r.generators_map_to_generators === true && return codomain(r)(map_entries(h, coordinates(x)))
-      v = images_of_generators(r)
-      return sum(h(a)*v[i] for (i, a) in coordinates(x); init=zero(codomain(r)))
+      return sum(h(b)*a[i] for (i, b) in coordinates(x); init=zero(codomain(r)))
     end
     function pr_func(x)
       @assert parent(x) === G

src/Modules/UngradedModules/FreeModElem.jl

@@ -216,14 +216,26 @@ function (==)(a::AbstractFreeModElem, b::AbstractFreeModElem)
   return a.coords == b.coords
 end
 
-function hash(a::AbstractFreeModElem, h::UInt)
+function hash(a::AbstractFreeModElem{<:MPolyElem}, h::UInt)
   b = 0xaa2ba4a32dd0b431 % UInt
   h = hash(typeof(a), h)
   h = hash(parent(a), h)
   h = hash(coordinates(a), h)
   return xor(h, b)
 end
 
+function hash(a::AbstractFreeModElem{<:MPolyQuoRingElem}, h::UInt)
+  simplify!(a)
+  b = 0xaa2ba4a32dd0b431 % UInt
+  h = hash(typeof(a), h)
+  h = hash(parent(a), h)
+  h = hash(coordinates(a), h)
+  return xor(h, b)
+end
+
+simplify(a::AbstractFreeModElem{<:MPolyQuoRingElem}) = return parent(a)(map_entries(simplify, coordinates(a)))
+simplify!(a::AbstractFreeModElem{<:MPolyQuoRingElem}) = simplify(a)
+
 function Base.deepcopy_internal(a::AbstractFreeModElem, dict::IdDict)
   return parent(a)(deepcopy_internal(coordinates(a), dict))
 end

src/Modules/UngradedModules/FreeModuleHom.jl

@@ -19,6 +19,7 @@ FreeModuleHom(F::AbstractFreeMod{T}, G::S, mat::MatElem{T}) where {T,S} = FreeMo
 
 img_gens(f::FreeModuleHom) = images_of_generators(f)
 images_of_generators(f::FreeModuleHom) = f.imgs_of_gens::Vector{elem_type(codomain(f))}
+image_of_generator(phi::FreeModuleHom, i::Int) = phi.imgs_of_gens[i]::elem_type(codomain(phi))
 base_ring_map(f::FreeModuleHom) = f.ring_map
 @attr Map function base_ring_map(f::FreeModuleHom{<:SubquoModule, <:ModuleFP, Nothing})
     return identity_map(base_ring(domain(f)))
@@ -239,7 +240,9 @@ hom(F::FreeMod, M::ModuleFP{T}, A::MatElem{T}, h::RingMapType; check::Bool=true)
 Return the identity map $id_M$.
 """
 function identity_map(M::ModuleFP)
-  return hom(M, M, gens(M), check=false)
+  phi = hom(M, M, gens(M), check=false)
+  phi.generators_map_to_generators = true
+  return phi
 end
 
 ### type getters in accordance with the `hom`-constructors
@@ -460,7 +463,9 @@ function kernel(h::FreeModuleHom)  #ONLY for free modules...
   R = base_ring(G)
   if ngens(G) == 0
     s = sub(G, gens(G), :module)
-    return s, hom(s, G, gens(G), check=false)
+    help = hom(s, G, gens(G), check=false)
+    help.generators_map_to_generators = true
+    return s, help
   end
   g = map(h, basis(G))
   if isa(codomain(h), SubquoModule)

src/Modules/UngradedModules/SubQuoHom.jl

@@ -81,6 +81,7 @@ function Base.show(io::IO, fmh::SubQuoHom{T1, T2, RingMapType}) where {T1 <: Abs
 end
 
 images_of_generators(phi::SubQuoHom) = phi.im::Vector{elem_type(codomain(phi))}
+image_of_generator(phi::SubQuoHom, i::Int) = phi.im[i]::elem_type(codomain(phi))
 
 ###################################################################
 
@@ -344,6 +345,15 @@ function Base.hash(f::ModuleFPHom{T}, h::UInt) where {S<:MPolyElem, T<:ModuleFP{
   return xor(h, b)
 end
 
+function Base.hash(f::ModuleFPHom, h::UInt)
+  b = 0x535bbdbb2bc54b46 % UInt
+  h = hash(typeof(f), h)
+  h = hash(domain(f), h)
+  h = hash(codomain(f), h)
+  # We can not assume that the images of generators 
+  # have a hash in general
+  return xor(h, b)
+end
 ###################################################################
 
 @doc raw"""
@@ -458,24 +468,17 @@ function image(f::SubQuoHom, a::SubquoModuleElem)
  #  f.generators_map_to_generators = images_of_generators(f) == gens(codomain(f))
  #end
   f.generators_map_to_generators === true && return codomain(f)(map_entries(base_ring_map(f), coordinates(a)))
-  # TODO matrix vector multiplication
-  i = zero(codomain(f))
-  b = coordinates(a)
-  for (p,v) = b
-    i += base_ring_map(f)(v)*f.im[p]
-  end
-  return i
+  return sum(b*image_of_generator(f, i) for (i, b) in coordinates(a); init=zero(codomain(f)))
 end
 
 function image(f::SubQuoHom{<:SubquoModule, <:ModuleFP, Nothing}, a::SubquoModuleElem)
   # TODO matrix vector multiplication
   @assert a.parent === domain(f)
-  if f.generators_map_to_generators === nothing
-    f.generators_map_to_generators = images_of_generators(f) == gens(codomain(f))
-  end
-  f.generators_map_to_generators && return codomain(f)(coordinates(a))
-  v = images_of_generators(f)
-  return sum(c*v[i] for (i, c) in coordinates(a); init=zero(codomain(f)))
+ #if f.generators_map_to_generators === nothing
+ #  f.generators_map_to_generators = images_of_generators(f) == gens(codomain(f))
+ #end
+  f.generators_map_to_generators === true && return codomain(f)(coordinates(a))
+  return sum(c*image_of_generator(f, i) for (i, c) in coordinates(a); init=zero(codomain(f)))
 end
 
 @doc raw"""
@@ -533,7 +536,9 @@ Additionally, if `I` denotes this object, return the inclusion map `I` $\to$ `co
 """
 function image(h::SubQuoHom)
   s = sub(codomain(h), images_of_generators(h), :module)
-  return s, hom(s, codomain(h), images_of_generators(h), check=false)
+  inc = hom(s, codomain(h), images_of_generators(h), check=false)
+  inc.generators_map_to_generators = true
+  return s, inc
 end
 
 @doc raw"""