diff --git a/engine-proto/engine.jl b/engine-proto/Engine.jl
similarity index 76%
rename from engine-proto/engine.jl
rename to engine-proto/Engine.jl
index 2f7294a..a632581 100644
--- a/engine-proto/engine.jl
+++ b/engine-proto/Engine.jl
@@ -1,3 +1,5 @@
+include("HittingSet.jl")
+
 module Engine
 
 export Construction, mprod
@@ -6,6 +8,24 @@ import Subscripts
 using LinearAlgebra
 using AbstractAlgebra
 using Groebner
+using ..HittingSet
+
+# --- commutative algebra ---
+
+# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
+# polynomial rings when the coefficients are integers. we use Groebner to extend
+# support to rationals and to finite fields of prime order
+Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
+  Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
+
+function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}}
+  leading = [exponent_vector(f, 1) for f in gens(I)]
+  targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading]
+  length(HittingSet.solve(HittingSetProblem(targets), maxdepth))
+end
+
+dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
+  length(gens(base_ring(I))) - codimension(I, maxdepth)
 
 # --- primitve elements ---
 
@@ -23,8 +43,6 @@ mutable struct Point{T} <: Element{T}
   ) where T = new(coords, vec, nothing)
 end
 
-##coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
-
 function buildvec!(pt::Point)
   coordring = parent(pt.coords[1])
   pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
@@ -43,8 +61,6 @@ mutable struct Sphere{T} <: Element{T}
   ) where T = new(coords, vec, rel)
 end
 
-##coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
-
 function buildvec!(sph::Sphere)
   coordring = parent(sph.coords[1])
   sph.vec = sph.coords
@@ -130,10 +146,6 @@ function realize(ctx::Construction{T}) where T
     end
   end
   
-  display(collect(elemenum))
-  display(coordnamelist)
-  println()
-  
   # construct coordinate ring
   coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
   
@@ -150,16 +162,14 @@ function realize(ctx::Construction{T}) where T
   # construct coordinate vectors
   for (_, elem) in elemenum
     buildvec!(elem)
-    display(elem.coords)
-    display(elem.vec)
-    println()
   end
   
   # turn relations into equations
-  vcat(
+  eqns = vcat(
     equation.(ctx.relations),
     [elem.rel for elem in ctx.elements if !isnothing(elem.rel)]
   )
+  Generic.Ideal(coordring, eqns)
 end
 
 end
@@ -172,22 +182,26 @@ a = Engine.Point{CoeffType}()
 s = Engine.Sphere{CoeffType}()
 a_on_s = Engine.LiesOn{CoeffType}(a, s)
 ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
-eqns_a_s = Engine.realize(ctx)
+ideal_a_s = Engine.realize(ctx)
+println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freeom")
 
 b = Engine.Point{CoeffType}()
 b_on_s = Engine.LiesOn{CoeffType}(b, s)
 Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
-eqns_ab_s = Engine.realize(ctx)
+ideal_ab_s = Engine.realize(ctx)
+println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freeom")
 
 spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 tangencies = [
   Engine.AlignsWithBy{CoeffType}(
     spheres[n],
     spheres[mod1(n+1, length(spheres))],
-    -1//1
+    CoeffType(-1//1)
   )
   for n in 1:3
 ]
-ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
\ No newline at end of file
+ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
+ideal_tan_sph = Engine.realize(ctx_tan_sph)
+println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freeom")
\ No newline at end of file
diff --git a/engine-proto/hitting-set.jl b/engine-proto/HittingSet.jl
similarity index 89%
rename from engine-proto/hitting-set.jl
rename to engine-proto/HittingSet.jl
index e9aacf4..347c4d2 100644
--- a/engine-proto/hitting-set.jl
+++ b/engine-proto/HittingSet.jl
@@ -1,13 +1,15 @@
 module HittingSet
 
+export HittingSetProblem, solve
+
 HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}}
 
-# `subsets` should be a collection of Set objects
-function HittingSetProblem(subsets, chosen = Set())
-  wholeset = union(subsets...)
+# `targets` should be a collection of Set objects
+function HittingSetProblem(targets, chosen = Set())
+  wholeset = union(targets...)
   T = eltype(wholeset)
   unsorted_moves = [
-    elt => Set(filter(s -> elt ∉ s, subsets))
+    elt => Set(filter(s -> elt ∉ s, targets))
     for elt in wholeset
   ]
   moves = sort(unsorted_moves, by = pair -> length(pair.second))
@@ -32,7 +34,6 @@ end
 
 function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T
   problems = Dict(pblm)
-  println(typeof(problems))
   while length(first(problems).first) < maxdepth
     subproblems = typeof(problems)()
     for (chosen, moves) in problems
@@ -56,7 +57,7 @@ end
 
 function test(n = 1)
   T = [Int64, Int64, Symbol, Symbol][n]
-  subsets = Set{T}.([
+  targets = Set{T}.([
     [
       [1, 3, 5],
       [2, 3, 4],
@@ -98,7 +99,7 @@ function test(n = 1)
       [:b, :z, :t14]
     ]
   ][n])
-  problem = HittingSetProblem(subsets)
+  problem = HittingSetProblem(targets)
   if isa(problem, HittingSetProblem{T})
     println("Correct type")
   else