diff --git a/docs/src/misc.md b/docs/src/misc.md
index 235997724a..3bb961a261 100644
--- a/docs/src/misc.md
+++ b/docs/src/misc.md
@@ -202,7 +202,7 @@ The available algorithms/normal forms are
 
 To select a normal form type for rings of type `NewRing`, implement the function
 ```julia
-Solve.matrix_normal_form_type(::NewRing) = Bla()
+Solve.matrix_normal_form_type(::Type{<:NewRing}) = Bla()
 ```
 where `Bla <: MatrixNormalFormTrait`.
 A new type trait can be added via
@@ -262,8 +262,8 @@ object.
 First of all, one needs to implement the function
 
 ```julia
-function Solve.solve_context_type(::NewRing)
-  return Solve.solve_context_type(::NormalFormTrait, elem_type(NewRing))
+function Solve.solve_context_type(T::Type{<:NewRing})
+  return Solve.solve_context_type(::NormalFormTrait, elem_type(T))
 end
 ```
 
diff --git a/src/Solve.jl b/src/Solve.jl
index de9147fe85..b3c206f5fa 100644
--- a/src/Solve.jl
+++ b/src/Solve.jl
@@ -88,22 +88,25 @@ struct LUTrait <: MatrixNormalFormTrait end # LU factoring for fields
 struct FFLUTrait <: MatrixNormalFormTrait end # "fraction free" LU factoring for fraction fields
 struct MatrixInterpolateTrait <: MatrixNormalFormTrait end # interpolate in fraction fields of polynomial rings
 
-function matrix_normal_form_type(R::Ring)
-  if is_domain_type(elem_type(R))
+function matrix_normal_form_type(T::Type{<:Ring})
+  if is_domain_type(T)
     return HermiteFormTrait()
   else
     return HowellFormTrait()
   end
 end
 
-matrix_normal_form_type(::Field) = RREFTrait()
+matrix_normal_form_type(::Type{<:Field}) = RREFTrait()
 
 # The fflu approach is the fastest over a fraction field (see benchmarks on PR 661)
-matrix_normal_form_type(::FracField) = FFLUTrait()
-matrix_normal_form_type(::AbstractAlgebra.Rationals{BigInt}) = FFLUTrait()
-matrix_normal_form_type(::FracField{T}) where {T <: PolyRingElem} = MatrixInterpolateTrait()
+matrix_normal_form_type(::Type{<:FracField}) = FFLUTrait()
+matrix_normal_form_type(::Type{<:AbstractAlgebra.Rationals{BigInt}}) = FFLUTrait()
+matrix_normal_form_type(::Type{<:FracField{T}}) where {T <: PolyRingElem} = MatrixInterpolateTrait()
 
-matrix_normal_form_type(A::MatElem) = matrix_normal_form_type(base_ring(A))
+matrix_normal_form_type(T::Type{<:MatElem}) = matrix_normal_form_type(base_ring_type(T))
+
+matrix_normal_form_type(x) = matrix_normal_form_type(typeof(x))
+matrix_normal_form_type(T::DataType) = throw(MethodError(matrix_normal_form_type, (T,)))
 
 ################################################################################
 #
@@ -207,29 +210,29 @@ function solve_init(NF::MatrixNormalFormTrait, A::MatElem)
   return solve_context_type(NF, base_ring(A))(A)
 end
 
-# For a ring R, the following signatures of `solve_context_type` need to be
+# For a ring R of type T, the following signatures of `solve_context_type` need to be
 # implemented:
-# 1) solve_context_type(R)
-# 2) solve_context_type(::MatrixNormalFormTrait, elem_type(R))
+# 1) solve_context_type(::Type{<:T})
+# 2) solve_context_type(::MatrixNormalFormTrait, elem_type(T))
 # Version 1 should pick a matrix_normal_form_type and call 2
 
-function solve_context_type(R::NCRing)
-  return solve_context_type(matrix_normal_form_type(R), elem_type(R))
+function solve_context_type(T::Type{<:NCRing})
+  return solve_context_type(matrix_normal_form_type(T), elem_type(T))
 end
 
-function solve_context_type(K::Field)
-  # matrix_normal_form_type(K) would be RREFTrait, but we want to use
+function solve_context_type(T::Type{<:Field})
+  # matrix_normal_form_type(T) would be RREFTrait, but we want to use
   # LU in solve contexts
-  return solve_context_type(LUTrait(), elem_type(K))
+  return solve_context_type(LUTrait(), elem_type(T))
 end
 
-function solve_context_type(K::Union{AbstractAlgebra.Rationals{BigInt}, FracField})
+function solve_context_type(T::Type{<:Union{AbstractAlgebra.Rationals{BigInt}, FracField}})
   # In this case, we use FFLU
-  return solve_context_type(FFLUTrait(), elem_type(K))
+  return solve_context_type(FFLUTrait(), elem_type(T))
 end
 
-function solve_context_type(A::MatElem)
-  return solve_context_type(base_ring(A))
+function solve_context_type(T::Type{<:MatElem})
+  return solve_context_type(base_ring_type(T))
 end
 
 function solve_context_type(NF::MatrixNormalFormTrait, ::Type{T}) where {T <: NCRingElement}
@@ -237,6 +240,10 @@ function solve_context_type(NF::MatrixNormalFormTrait, ::Type{T}) where {T <: NC
   return SolveCtx{T, typeof(NF), MatType, MatType, LazyTransposeMatElem{T, MatType}}
 end
 
+function solve_context_type(NF::MatrixNormalFormTrait, T::Type{<:MatElem})
+  return solve_context_type(NF, base_ring_type(T))
+end
+
 function solve_context_type(::FFLUTrait, ::Type{T}) where {T <: NCRingElement}
   # We assume that the ring in question is a fraction field and have to get the
   # type of "integral" matrices, that is, matrices over the base ring of this
@@ -246,13 +253,15 @@ function solve_context_type(::FFLUTrait, ::Type{T}) where {T <: NCRingElement}
   return SolveCtx{T, FFLUTrait, dense_matrix_type(T), IntMatT, IntMatT}
 end
 
-solve_context_type(NF::MatrixNormalFormTrait, ::T) where {T <: NCRingElement} = solve_context_type(NF, T)
 solve_context_type(NF::MatrixNormalFormTrait, ::Type{T}) where {T <: NCRing} = solve_context_type(NF, elem_type(T))
-solve_context_type(NF::MatrixNormalFormTrait, ::T) where {T <: NCRing} = solve_context_type(NF, elem_type(T))
-solve_context_type(NF::MatrixNormalFormTrait, ::Type{<: MatElem{T}}) where T = solve_context_type(NF, T)
-solve_context_type(NF::MatrixNormalFormTrait, ::MatElem{T}) where T = solve_context_type(NF, T)
 
-matrix_normal_form_type(C::SolveCtx{T, NF}) where {T, NF} = NF()
+solve_context_type(x) = solve_context_type(typeof(x))
+solve_context_type(NF::MatrixNormalFormTrait, x) = solve_context_type(NF, typeof(x))
+solve_context_type(T::DataType) = throw(MethodError(solve_context_type, (T,)))
+solve_context_type(NF::MatrixNormalFormTrait, T::DataType) = throw(MethodError(solve_context_type, (NF, T)))
+
+
+matrix_normal_form_type(::Type{<:SolveCtx{T, NF}}) where {T, NF} = NF()
 
 matrix(C::SolveCtx) = C.A
 
diff --git a/test/Solve-test.jl b/test/Solve-test.jl
index 849d3d37f7..93446e92e2 100644
--- a/test/Solve-test.jl
+++ b/test/Solve-test.jl
@@ -103,13 +103,17 @@ end
 
   if is_default
     C = solve_init(M)
+    @test AbstractAlgebra.Solve.matrix_normal_form_type(typeof(C)) === NFTrait()
     @test AbstractAlgebra.Solve.matrix_normal_form_type(C) === NFTrait()
+    @test C isa AbstractAlgebra.solve_context_type(typeof(R))
     @test C isa AbstractAlgebra.solve_context_type(R)
+    @test C isa AbstractAlgebra.solve_context_type(typeof(M))
     @test C isa AbstractAlgebra.solve_context_type(M)
   end
 
   C = solve_init(NFTrait(), M)
 
+  @test AbstractAlgebra.Solve.matrix_normal_form_type(typeof(C)) === NFTrait()
   @test AbstractAlgebra.Solve.matrix_normal_form_type(C) === NFTrait()
   @test C isa AbstractAlgebra.solve_context_type(NFTrait(), elem_type(R))
   @test C isa AbstractAlgebra.solve_context_type(NFTrait(), R(1))