diff --git a/Project.toml b/Project.toml
index 2ed20dd34..18bb57a35 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRulesCore"
 uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
-version = "0.5.0"
+version = "0.5.1"
 
 [compat]
 julia = "^1.0"
diff --git a/src/differential_arithmetic.jl b/src/differential_arithmetic.jl
index 2d4434a93..985f1204f 100644
--- a/src/differential_arithmetic.jl
+++ b/src/differential_arithmetic.jl
@@ -32,17 +32,16 @@ Base.:*(::DoesNotExist, ::Zero) = Zero()
 Base.:*(::Zero, ::DoesNotExist) = Zero()
 
 
-Base.:+(::Zero, b::Zero) = Zero()
+Base.:+(::Zero, ::Zero) = Zero()
 Base.:*(::Zero, ::Zero) = Zero()
 for T in (:One, :AbstractThunk, :Any)
     @eval Base.:+(::Zero, b::$T) = b
     @eval Base.:+(a::$T, ::Zero) = a
 
-    @eval Base.:*(::Zero, ::$T) = Zero()
-    @eval Base.:*(::$T, ::Zero) = Zero()
+    @eval Base.:*(::Zero, x::$T) = zero(x)
+    @eval Base.:*(x::$T, ::Zero) = zero(x)
 end
 
-
 Base.:+(a::One, b::One) = extern(a) + extern(b)
 Base.:*(::One, ::One) = One()
 for T in (:AbstractThunk, :Any)
diff --git a/src/differentials/zero.jl b/src/differentials/zero.jl
index 3e5312a72..2249a0b95 100644
--- a/src/differentials/zero.jl
+++ b/src/differentials/zero.jl
@@ -11,3 +11,5 @@ Base.Broadcast.broadcastable(::Zero) = Ref(Zero())
 
 Base.iterate(x::Zero) = (x, nothing)
 Base.iterate(::Zero, ::Any) = nothing
+
+Base.zero(::AbstractDifferential) = Zero()
diff --git a/test/differentials/zero.jl b/test/differentials/zero.jl
index 54c16367c..30f89a76f 100644
--- a/test/differentials/zero.jl
+++ b/test/differentials/zero.jl
@@ -1,15 +1,19 @@
 @testset "Zero" begin
     z = Zero()
     @test extern(z) === false
-    @test z + z == z
-    @test z + 1 == 1
-    @test 1 + z == 1
-    @test z * z == z
-    @test z * 1 == z
-    @test 1 * z == z
+    @test z + z === z
+    @test z + 1 === 1
+    @test 1 + z === 1
+    @test z * z === z
+    @test z * 1 === 0
+    @test 1 * z === 0
     for x in z
         @test x === z
     end
     @test broadcastable(z) isa Ref{Zero}
-    @test conj(z) == z
+    @test conj(z) === z
+    @test zero(@thunk(3)) === z
+    @test zero(One()) === z
+    @test zero(DoesNotExist()) === z
+    @test zero(Composite{Tuple{Int,Int}}((1, 2))) === z
 end
diff --git a/test/rules.jl b/test/rules.jl
index f280ebc59..1845f5739 100644
--- a/test/rules.jl
+++ b/test/rules.jl
@@ -8,6 +8,9 @@ cool(x, y) = x + y + 1
 dummy_identity(x) = x
 @scalar_rule(dummy_identity(x), One())
 
+nice(x) = 1
+@scalar_rule(nice(x), Zero())
+
 #######
 
 _second(t) = Base.tuple_type_head(Base.tuple_type_tail(t))
@@ -31,11 +34,16 @@ _second(t) = Base.tuple_type_head(Base.tuple_type_tail(t))
     @test cool_methods == only_methods
 
     frx, cool_pushforward = frule(cool, 1, dself, 1)
-    @test frx == 2
-    @test cool_pushforward == 1
+    @test frx === 2
+    @test cool_pushforward === 1
     rrx, cool_pullback = rrule(cool, 1)
     self, rr1 = cool_pullback(1)
-    @test self == NO_FIELDS
-    @test rrx == 2
-    @test rr1 == 1
+    @test self === NO_FIELDS
+    @test rrx === 2
+    @test rr1 === 1
+
+    frx, nice_pushforward = frule(nice, 1, dself, 1)
+    @test nice_pushforward === 0
+    rrx, nice_pullback = rrule(nice, 1)
+    @test (NO_FIELDS, 0) === nice_pullback(1)
 end