diff --git a/src/Modules/ModuleTypes.jl b/src/Modules/ModuleTypes.jl
index cdbfe1ed9571..818929471599 100644
--- a/src/Modules/ModuleTypes.jl
+++ b/src/Modules/ModuleTypes.jl
@@ -82,6 +82,7 @@ option is set in suitable functions.
   n::Int
   S::Vector{Symbol}
   d::Union{Vector{FinGenAbGroupElem}, Nothing}
+  default_ordering::ModuleOrdering
 
   # We register the incoming and outgoing natural morphisms.
   # This must be done in a way that objects can be collected by the 
diff --git a/src/Modules/UngradedModules/Methods.jl b/src/Modules/UngradedModules/Methods.jl
index d86acd1ade09..92d66bb355f8 100644
--- a/src/Modules/UngradedModules/Methods.jl
+++ b/src/Modules/UngradedModules/Methods.jl
@@ -229,10 +229,14 @@ ring_map(f::SubQuoHom{<:AbstractFreeMod, <:ModuleFP, Nothing}) = nothing
 ring_map(f::SubQuoHom) = f.ring_map
 
 function default_ordering(F::FreeMod)
-  if iszero(F)
-    return default_ordering(base_ring(F))*ModuleOrdering(F, Orderings.ModOrdering(Vector{Int}(), :lex))
+  if !isdefined(F, :default_ordering)
+    if iszero(F)
+      F.default_ordering = default_ordering(base_ring(F))*ModuleOrdering(F, Orderings.ModOrdering(Vector{Int}(), :lex))
+    else
+      F.default_ordering = default_ordering(base_ring(F))*lex(gens(F))
+    end
   end
-  return default_ordering(base_ring(F))*lex(gens(F))
+  return F.default_ordering::ModuleOrdering{typeof(F)}
 end
 
 ##############################
diff --git a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
index ca8ae8494564..c53cb41fbe81 100644
--- a/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
+++ b/src/Modules/UngradedModules/SubModuleOfFreeModule.jl
@@ -143,6 +143,7 @@ end
 Get the default ordering of `M`.
 """
 function default_ordering(M::SubModuleOfFreeModule)
+  return default_ordering(ambient_free_module(M))
   if !isdefined(M, :default_ordering)
     ord = default_ordering(ambient_free_module(M))
     set_default_ordering!(M, ord)
@@ -166,7 +167,14 @@ end
 Compute a standard basis of `submod` with respect to the given `odering``.
 The return type is `ModuleGens`.
 """
-function standard_basis(submod::SubModuleOfFreeModule; ordering::ModuleOrdering = default_ordering(submod))
+function standard_basis(submod::SubModuleOfFreeModule; ordering::Union{ModuleOrdering, Nothing} = default_ordering(submod))
+  # This is to circumvent hashing of the ordering in the obviously avoidable cases
+  if ordering===default_ordering(submod)
+    for (ord, gb) in submod.groebner_basis
+      ord === ordering && return gb
+    end
+  end
+    
   @req is_exact_type(elem_type(base_ring(submod))) "This functionality is only supported over exact fields."
   gb = get!(submod.groebner_basis, ordering) do
     return compute_standard_basis(submod, ordering)