diff --git a/src/Map.jl b/src/Map.jl
index a5255261b..1e8199e81 100644
--- a/src/Map.jl
+++ b/src/Map.jl
@@ -17,6 +17,10 @@ function domain end
 function codomain end
 function image_fn end
 
+function coimage(h::Map)
+  return quo(domain(h), kernel(h)[1])
+end
+
 function check_composable(a::Map, b::Map)
    codomain(a) !== domain(b) && error("Incompatible maps")
 end
diff --git a/src/Matrix.jl b/src/Matrix.jl
index 00d80d5c7..f129cd9e8 100644
--- a/src/Matrix.jl
+++ b/src/Matrix.jl
@@ -502,6 +502,9 @@ end
 Base.IteratorSize(::Type{<:MatrixElem}) = Base.HasShape{2}()
 Base.IteratorEltype(::Type{<:MatrixElem}) = Base.HasEltype() # default
 
+Base.pairs(M::MatElem) = Base.pairs(IndexCartesian(), M)
+Base.pairs(::IndexCartesian, M::MatElem) = Base.Iterators.Pairs(M, CartesianIndices(axes(M)))
+
 ###############################################################################
 #
 #   Block replacement
diff --git a/src/Module.jl b/src/Module.jl
index 4a6d8e176..138e7d1da 100644
--- a/src/Module.jl
+++ b/src/Module.jl
@@ -10,6 +10,8 @@
 #
 ###############################################################################
 
+Base.eltype(M::FPModule{T}) where T <: FinFieldElem = elem_type(M)
+
 function zero(M::FPModule{T}) where T <: RingElement
    R = base_ring(M)
    return M(zero_matrix(R, 1, ngens(M)))
@@ -27,6 +29,21 @@ function check_parent(M::FPModuleElem{T}, N::FPModuleElem{T}) where T <: RingEle
    parent(M) !== parent(N) && error("Incompatible modules")
 end
 
+is_finite(M::FPModule{<:FinFieldElem}) = true
+
+function is_sub_with_data(M::FPModule{T}, N::FPModule{T}) where T <: RingElement
+  fl = is_submodule(N, M)
+  if fl
+    return fl, hom(M, N, elem_type(N)[N(m) for m = gens(M)])
+  else
+    return fl, hom(M, N, elem_type(N)[zero(N) for m = gens(M)])
+  end
+end
+
+Base.issubset(M::FPModule{T}, N::FPModule{T}) where T <: RingElement = is_submodule(M, N)
+
+order(M::FPModule{<:FinFieldElem}) = order(base_ring(M))^dim(M)
+
 ###############################################################################
 #
 #   Unary operators
@@ -300,3 +317,33 @@ function rand(rng::AbstractRNG, M::FPModule{T}, vals...) where T <: RingElement
 end
 
 rand(M::FPModule, vals...) = rand(Random.GLOBAL_RNG, M, vals...)
+
+###############################################################################
+#
+#   Iteration
+#
+###############################################################################
+
+Base.length(M::FPModule{T}) where T <: FinFieldElem = Int(order(M))
+
+function Base.iterate(M::FPModule{T}) where T <: FinFieldElem
+  k = base_ring(M)
+  if dim(M) == 0
+    return zero(M), iterate([1])
+  end
+  p = Base.Iterators.ProductIterator(Tuple([k for i=1:dim(M)]))
+  f = iterate(p)
+  return M(elem_type(k)[f[1][i] for i=1:dim(M)]), (f[2], p)
+end
+
+function Base.iterate(M::FPModule{T}, st::Tuple{<:Tuple, <:Base.Iterators.ProductIterator}) where T <: FinFieldElem
+  n = iterate(st[2], st[1])
+  if n === nothing
+    return n
+  end
+  return M(elem_type(base_ring(M))[n[1][i] for i=1:dim(M)]), (n[2], st[2])
+end
+
+function Base.iterate(::FPModule{<:FinFieldElem}, ::Tuple{Int64, Int64})
+  return nothing
+end
diff --git a/src/exports.jl b/src/exports.jl
index 238ad3655..504ff4d2c 100644
--- a/src/exports.jl
+++ b/src/exports.jl
@@ -273,6 +273,7 @@ export is_even
 export is_exact_type
 export is_finite
 export is_finite_order
+export is_free
 export is_gen
 export is_hermitian
 export is_hessenberg
@@ -308,6 +309,7 @@ export is_square_with_sqrt
 export is_squarefree
 export is_submodule
 export is_subset
+export is_sub_with_data
 export is_symmetric
 export is_term
 export is_term_recursive
diff --git a/src/generic/DirectSum.jl b/src/generic/DirectSum.jl
index 1500d41fc..82e672ac3 100644
--- a/src/generic/DirectSum.jl
+++ b/src/generic/DirectSum.jl
@@ -20,6 +20,8 @@ base_ring(N::DirectSumModule{T}) where T <: RingElement = base_ring(N.m[1])
 
 base_ring(v::DirectSumModuleElem{T}) where T <: RingElement = base_ring(v.parent)
 
+dim(M::DirectSumModule{<:FieldElem}) = sum(dim(x) for x = M.m)
+
 number_of_generators(N::DirectSumModule{T}) where T <: RingElement = sum(ngens(M) for M in N.m)
 
 gens(N::DirectSumModule{T}) where T <: RingElement = [gen(N, i) for i = 1:ngens(N)]
@@ -247,6 +249,11 @@ function ModuleHomomorphism(D::DirectSumModule{T}, A::DirectSumModule{T}, m::Mat
    return ModuleHomomorphism(D, A, transpose(hvcat(Tuple([length(SD) for i = 1:length(SA)]), map(x->transpose(x.matrix), m)...)))
 end
 
+function AbstractAlgebra.hom(M::DirectSumModule{T}, N::DirectSumModule{T}, mp::Vector{ModuleHomomorphism{T}}) where T
+  @assert length(M.m) == length(mp) == length(N.m)
+  return hom(M, N, cat(map(matrix, mp)..., dims = (1,2)))
+end
+
 function AbstractAlgebra.hom(A::AbstractAlgebra.Generic.DirectSumModule{T}, B::AbstractAlgebra.Generic.DirectSumModule{T}, M::Matrix{<:Map{<:AbstractAlgebra.FPModule{T}, <:AbstractAlgebra.FPModule{T}}}) where {T}
   pro = canonical_projections(A)
   im = canonical_injections(B)
diff --git a/src/generic/FreeModule.jl b/src/generic/FreeModule.jl
index ab20cec12..79f8403df 100644
--- a/src/generic/FreeModule.jl
+++ b/src/generic/FreeModule.jl
@@ -29,6 +29,8 @@ function check_parent(m1::FreeModuleElem{T}, m2::FreeModuleElem{T}) where T <: U
     parent(m1) !== parent(m2) && error("Incompatible free modules")
 end
 
+is_free(M::FreeModule) = true
+
 @doc raw"""
     rank(M::FreeModule{T}) where T <: Union{RingElement, NCRingElem}
 
diff --git a/src/generic/Map.jl b/src/generic/Map.jl
index 38dbb762f..14dbb220f 100644
--- a/src/generic/Map.jl
+++ b/src/generic/Map.jl
@@ -91,6 +91,8 @@ end
 
 Base.inv(f::AbstractAlgebra.Map(AbstractAlgebra.IdentityMap)) = f
 
+AbstractAlgebra.matrix(phi::IdentityMap{<:AbstractAlgebra.FPModule}) = identity_matrix(base_ring(domain(phi)), dim(domain(phi)))
+
 ################################################################################
 #
 #  FunctionalMap
diff --git a/src/generic/Misc/Poly.jl b/src/generic/Misc/Poly.jl
index 7d323d82c..3d50943a2 100644
--- a/src/generic/Misc/Poly.jl
+++ b/src/generic/Misc/Poly.jl
@@ -75,6 +75,11 @@ function roots(R::Field, f::PolyRingElem)
     return roots(f1)
 end
 
+function roots(a::FinFieldElem, i::Int)
+  _, x = polynomial_ring(parent(a), cached = false)
+  return roots(x^i-a)
+end
+
 function sturm_sequence(f::PolyRingElem{<:FieldElem})
     g = f
     h = derivative(g)
diff --git a/src/generic/ModuleHomomorphism.jl b/src/generic/ModuleHomomorphism.jl
index 136e53ba8..920cad8fe 100644
--- a/src/generic/ModuleHomomorphism.jl
+++ b/src/generic/ModuleHomomorphism.jl
@@ -22,6 +22,44 @@ inverse_mat(f::Map(ModuleIsomorphism)) = f.inverse_matrix
 
 inverse_image_fn(f::Map(ModuleIsomorphism)) = f.inverse_image_fn
 
+###############################################################################
+#
+#   Unary operators
+#
+###############################################################################
+
+Base.:-(a::ModuleHomomorphism) = hom(domain(a), codomain(a), -matrix(a))
+
+###############################################################################
+#
+#   Binary operators
+#
+###############################################################################
+
+Base.:*(a::T, b::ModuleHomomorphism{T}) where {T} = hom(domain(b), codomain(b), a * matrix(b))
+Base.:*(a::T, b::ModuleIsomorphism{T}) where {T} = hom(domain(b), codomain(b), a * matrix(b))
+Base.:+(a::ModuleHomomorphism, b::ModuleHomomorphism) = hom(domain(a), codomain(a), matrix(a) + matrix(b))
+Base.:-(a::ModuleHomomorphism, b::ModuleHomomorphism) = hom(domain(a), codomain(a), matrix(a) - matrix(b))
+
+###############################################################################
+#
+#   Comparison
+#
+###############################################################################
+
+function Base.:(==)(a::Union{ModuleHomomorphism, ModuleIsomorphism}, b::Union{ModuleHomomorphism, ModuleIsomorphism})
+  domain(a) === domain(b) || return false
+  codomain(a) === codomain(b) || return false
+  return matrix(a) == matrix(b)
+end
+
+function Base.hash(a::Union{ModuleHomomorphism, ModuleIsomorphism}, h::UInt)
+  h = hash(domain(a), h)
+  h = hash(codomain(a), h)
+  h = hash(matrix(a), h)
+  return h
+end
+
 ###############################################################################
 #
 #   String I/O
@@ -68,6 +106,10 @@ function Base.inv(f::Map(ModuleIsomorphism))
    return ModuleIsomorphism{T}(codomain(f), domain(f), inverse_mat(f), matrix(f))
 end
 
+function Base.inv(f::ModuleHomomorphism)
+  return hom(codomain(f), domain(f), inv(matrix(f)))
+end
+
 ###############################################################################
 #
 #   Call overload
@@ -139,3 +181,14 @@ function ModuleIsomorphism(M1::AbstractAlgebra.FPModule{T},
    end
    return ModuleIsomorphism{T}(M1, M2, M, M_inv)
 end
+
+function hom(V::AbstractAlgebra.Module, W::AbstractAlgebra.Module, v::Vector{<:ModuleElem}; check::Bool = true)
+  if ngens(V) == 0
+    return ModuleHomomorphism(V, W, zero_matrix(base_ring(V), ngens(V), ngens(W)))
+  end
+  return ModuleHomomorphism(V, W, reduce(vcat, [x.v for x = v]))
+end
+
+function hom(V::AbstractAlgebra.Module, W::AbstractAlgebra.Module, v::MatElem; check::Bool = true)
+  return ModuleHomomorphism(V, W, v)
+end
diff --git a/src/generic/exports.jl b/src/generic/exports.jl
index 6ea87f6b6..517cbfd10 100644
--- a/src/generic/exports.jl
+++ b/src/generic/exports.jl
@@ -47,6 +47,7 @@ export inverse_mat
 export invmod
 export is_compatible
 export is_divisible_by
+export is_free
 export is_homogeneous
 export is_power
 export is_rimhook