diff --git a/src/NCRings.jl b/src/NCRings.jl
index 24a9c40ede..3b082fab3e 100644
--- a/src/NCRings.jl
+++ b/src/NCRings.jl
@@ -83,13 +83,17 @@ end
 
 *(x::NCRingElement, y::NCRingElem) = parent(y)(x)*y
 
+# general fallback for comparison of elements via promotion.
+# This is different from Julia's existing promotion logic because
+# it takes parents into account
 function ==(x::NCRingElem, y::NCRingElem)
-   fl, u, v = try_promote(x, y)
-   if fl
-     return u == v
-   else
-     return false
-   end
+  # avoid infinite recursion: we only get here if a ring type "forgot" to
+  # implement ==, so only do something if x and y have different type
+  if typeof(x) !== typeof(y)
+    fl, u, v = try_promote(x, y)
+    fl && return u == v
+  end
+  throw(NotImplementedError(:(==), x, y))
  end
  
 ==(x::NCRingElem, y::NCRingElement) = x == parent(x)(y)
diff --git a/test/generic/AbsSeries-test.jl b/test/generic/AbsSeries-test.jl
index 67de078a50..fabf56cd37 100644
--- a/test/generic/AbsSeries-test.jl
+++ b/test/generic/AbsSeries-test.jl
@@ -90,13 +90,6 @@ end
 
    @test isa(l, Generic.AbsSeries)
 
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
-
    R, x = power_series_ring(ZZ, 30, "x", model=:capped_absolute)
    S, x = power_series_ring(ZZ, 30, "x", model=:capped_absolute)
    @test R === S
diff --git a/test/generic/FactoredFraction-test.jl b/test/generic/FactoredFraction-test.jl
index 1944a0d461..3e70bf9d35 100644
--- a/test/generic/FactoredFraction-test.jl
+++ b/test/generic/FactoredFraction-test.jl
@@ -218,12 +218,6 @@ end
     TT = factored_fraction_field(polynomial_ring(QQ, "x")[1])
     a = TT(1)
     b = T(2)
-
-    @test a in [a, b]
-    @test a in [b, a]
-    @test !(a in [b])
-    @test a in keys(Dict(a => 1))
-    @test !(b in keys(Dict(a => 1)))
 end
 
 @testset "Generic.FactoredFracFieldElem.printing" begin
diff --git a/test/generic/Fraction-test.jl b/test/generic/Fraction-test.jl
index bdfe96bcb1..21b559adf1 100644
--- a/test/generic/Fraction-test.jl
+++ b/test/generic/Fraction-test.jl
@@ -46,12 +46,6 @@
    b = T(2)
    @test is_perfect(TT)
 
-   @test a in [a, b]
-   @test a in [b, a]
-   @test !(a in [b])
-   @test a in keys(Dict(a => 1))
-   @test !(b in keys(Dict(a => 1)))
-
    # trivial rings can not be fields
    R = residue_ring(ZZ, 1)[1]
    @test is_trivial(R)
diff --git a/test/generic/FreeAssociativeAlgebra-test.jl b/test/generic/FreeAssociativeAlgebra-test.jl
index dad6935b0e..ec18a7d638 100644
--- a/test/generic/FreeAssociativeAlgebra-test.jl
+++ b/test/generic/FreeAssociativeAlgebra-test.jl
@@ -125,14 +125,6 @@
 
       @test collect(exponent_words(varlist[1] + 1)) == [Int[1], Int[]]
       @test isone(varlist[1]^0)
-
-      _, varlist = polynomial_ring(QQ, var_names)
-      y = varlist[1]
-      @test x in [x, y]
-      @test x in [y, x]
-      @test !(x in [y])
-      @test x in keys(Dict(x => 1))
-      @test !(y in keys(Dict(x => 1)))
    end
 end
 
diff --git a/test/generic/FunctionField-test.jl b/test/generic/FunctionField-test.jl
index 3afcb3219e..f617b675e1 100644
--- a/test/generic/FunctionField-test.jl
+++ b/test/generic/FunctionField-test.jl
@@ -74,13 +74,6 @@ P2 = [(x2 + 1)*z2 + (x2 + 2), z2 + (x2 + 1)//(x2 + 2), z2^2 + 3z2 + 1,
    @test !is_perfect(S2)
 
    @test var(S1) == :y1
-
-   @test y1 in [y1, y2]
-   @test y1 in [y2, y1]
-   @test !(y1 in [y2])
-
-   @test y1 in keys(Dict(y1 => 1))
-   @test !(y2 in keys(Dict(y1 => 1)))
 end
 
 @testset "Generic.FunctionField.printing" begin
diff --git a/test/generic/LaurentSeries-test.jl b/test/generic/LaurentSeries-test.jl
index c3fa19df1c..674f4e2672 100644
--- a/test/generic/LaurentSeries-test.jl
+++ b/test/generic/LaurentSeries-test.jl
@@ -104,13 +104,6 @@ end
    l = T(t)
 
    @test isa(l, Generic.LaurentSeriesElem)
-
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
    
    R, x = laurent_series_ring(ZZ, 30, "x")
    RR, x = laurent_series_ring(ZZ, 30, "x")
diff --git a/test/generic/MPoly-test.jl b/test/generic/MPoly-test.jl
index a758dcdc91..84e52e6cdd 100644
--- a/test/generic/MPoly-test.jl
+++ b/test/generic/MPoly-test.jl
@@ -83,14 +83,6 @@
       f3 = finish(C)
 
       @test f1 == f3
-
-      _, varlist = polynomial_ring(QQ, var_names)
-      y = varlist[1]
-      @test x in [x, y]
-      @test x in [y, x]
-      @test !(x in [y])
-      @test x in keys(Dict(x => 1))
-      @test !(y in keys(Dict(x => 1)))
    end
 
    R1, (x, y) = polynomial_ring(QQ, ["x", "y"])
diff --git a/test/generic/Matrix-test.jl b/test/generic/Matrix-test.jl
index 59fa2f18a4..5e367264d6 100644
--- a/test/generic/Matrix-test.jl
+++ b/test/generic/Matrix-test.jl
@@ -308,9 +308,23 @@ end
    @test D4[5, 5] == R(5)
    @test D4 isa Generic.MatSpaceElem{elem_type(R)}
 
+   # test comparison of matrices over different ring, and of different sizes
    x = zero_matrix(R, 2, 2)
    y = zero_matrix(ZZ, 2, 3)
 
+   @test x != y
+   @test x in [x, y]
+   @test x in [y, x]
+   @test !(x in [y])
+
+   @test x in keys(Dict(x => 1))
+   @test !(y in keys(Dict(x => 1)))
+
+   # test comparison of matrices over same ring, but of different sizes
+   x = zero_matrix(ZZ, 2, 2)
+   y = zero_matrix(ZZ, 2, 3)
+
+   @test x != y
    @test x in [x, y]
    @test x in [y, x]
    @test !(x in [y])
diff --git a/test/generic/Poly-test.jl b/test/generic/Poly-test.jl
index 7a8bae24d4..51a533d3a5 100644
--- a/test/generic/Poly-test.jl
+++ b/test/generic/Poly-test.jl
@@ -95,13 +95,6 @@ end
       n = S([ZZ(1), ZZ(2), ZZ(3)])
 
       @test isa(n, PolyRingElem)
-
-      @test x in [x, y]
-      @test x in [y, x]
-      @test !(x in [y])
-
-      @test x in keys(Dict(x => 1))
-      @test !(y in keys(Dict(x => 1)))
    end
 
    @testset "Generic.Poly.constructors.varname" begin
diff --git a/test/generic/PuiseuxSeries-test.jl b/test/generic/PuiseuxSeries-test.jl
index 930946a89c..b9c1dcde0e 100644
--- a/test/generic/PuiseuxSeries-test.jl
+++ b/test/generic/PuiseuxSeries-test.jl
@@ -98,13 +98,6 @@ end
    l = T(t)
 
    @test isa(l, Generic.PuiseuxSeriesElem)
-
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
 end
 
 @testset "Generic.PuiseuxSeries.printing" begin
diff --git a/test/generic/RationalFunctionField-test.jl b/test/generic/RationalFunctionField-test.jl
index ed9827cb03..8d3b114609 100644
--- a/test/generic/RationalFunctionField-test.jl
+++ b/test/generic/RationalFunctionField-test.jl
@@ -40,12 +40,6 @@ end
    a = TT(1)
    b = T(2)
 
-   @test a in [a, b]
-   @test a in [b, a]
-   @test !(a in [b])
-   @test a in keys(Dict(a => 1))
-   @test !(b in keys(Dict(a => 1)))
-
    # Multivariate
 
    T, (x, y) = rational_function_field(QQ, ["x", "y"])
diff --git a/test/generic/RelSeries-test.jl b/test/generic/RelSeries-test.jl
index edb03b32f7..a7afc80297 100644
--- a/test/generic/RelSeries-test.jl
+++ b/test/generic/RelSeries-test.jl
@@ -99,13 +99,6 @@ end
 
    @test isa(l, Generic.RelSeries)
 
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
-
    @test_throws DomainError O(0+O(x^0))
 
    @test !occursin("\n", sprint(show, T))
diff --git a/test/generic/Residue-test.jl b/test/generic/Residue-test.jl
index 06054de76e..ebf94f9bcf 100644
--- a/test/generic/Residue-test.jl
+++ b/test/generic/Residue-test.jl
@@ -105,16 +105,6 @@ end
    k = T(1)
 
    @test isa(k, EuclideanRingResidueRingElem)
-
-   S, = Generic.residue_ring(B, 164538890)
-   x = R(1)
-   y = S(1)
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
 end
 
 @testset "EuclideanRingResidueRingElem.is_trivial" begin
diff --git a/test/generic/ResidueField-test.jl b/test/generic/ResidueField-test.jl
index 5429328468..e306a46cff 100644
--- a/test/generic/ResidueField-test.jl
+++ b/test/generic/ResidueField-test.jl
@@ -58,16 +58,6 @@
 
    @test isa(g, EuclideanRingResidueFieldElem)
 
-   S, = Generic.residue_ring(B, 2)
-   x = R(1)
-   y = S(1)
-   @test x in [x, y]
-   @test x in [y, x]
-   @test !(x in [y])
-
-   @test x in keys(Dict(x => 1))
-   @test !(y in keys(Dict(x => 1)))
-
    # trivial rings can not be fields
    @test_throws ArgumentError residue_field(ZZ, 1)
    R = residue_ring(ZZ, 1)[1]
diff --git a/test/generic/UnivPoly-test.jl b/test/generic/UnivPoly-test.jl
index 9c4249741d..bc3e891430 100644
--- a/test/generic/UnivPoly-test.jl
+++ b/test/generic/UnivPoly-test.jl
@@ -75,15 +75,6 @@
 
          @test f1 == f3
 
-         @test x in [x, y]
-         @test x in [y, x]
-         @test y in [y, z]
-         @test !(x in [y])
-         @test !(y in [x])
-         @test x in keys(Dict(x => 1))
-         @test !(y in keys(Dict(x => 1)))
-         @test !(y in keys(Dict(z => 1)))
-
          S2, x = universal_polynomial_ring(R, :x => 1:3; internal_ordering=ord)
 
          @test length(x) == 3