diff --git a/src/algebra_elts.jl b/src/algebra_elts.jl
index 8644903..e2e160b 100644
--- a/src/algebra_elts.jl
+++ b/src/algebra_elts.jl
@@ -28,9 +28,14 @@ end
 function LinearAlgebra.norm(a::AlgebraElement, p::Real)
     return LinearAlgebra.norm(coeffs(a), p)
 end
-function LinearAlgebra.dot(a::AlgebraElement, v::AbstractVector)
-    return LinearAlgebra.dot(coeffs(a), v)
+
+function LinearAlgebra.dot(a::AlgebraElement, b::AlgebraElement)
+    return LinearAlgebra.dot(coeffs(a), coeffs(b))
+end
+
+function LinearAlgebra.dot(w::AbstractVector, b::AlgebraElement)
+    return LinearAlgebra.dot(w, coeffs(b))
 end
-function LinearAlgebra.dot(v::AbstractVector, a::AlgebraElement)
-    return LinearAlgebra.dot(a, v)
+function LinearAlgebra.dot(a::AlgebraElement, w::AbstractVector)
+    return LinearAlgebra.dot(coeffs(a), w)
 end
diff --git a/src/coefficients.jl b/src/coefficients.jl
index df104f9..735be39 100644
--- a/src/coefficients.jl
+++ b/src/coefficients.jl
@@ -92,14 +92,34 @@ end
 end
 
 function LinearAlgebra.norm(sc::AbstractCoefficients, p::Real)
-    isempty(keys(sc)) && return (0^p)^1 / p
-    return sum(v^p for v in values(sc))^1 / p
+    isempty(values(sc)) && return (0^p)^(1 / p)
+    return sum(abs(v)^p for v in values(sc))^(1 / p)
+end
+
+function LinearAlgebra.dot(ac::AbstractCoefficients, bc::AbstractCoefficients)
+    if isempty(values(ac)) || isempty(values(bc))
+        return zero(Base._return_type(*, Tuple{valtype(ac),valtype(bc)}))
+    else
+        return sum(c * star(bc[i]) for (i, c) in nonzero_pairs(ac))
+    end
+end
+
+function LinearAlgebra.dot(w::AbstractVector, ac::AbstractCoefficients)
+    @assert key_type(ac) <: Integer
+    if isempty(values(ac))
+        return zero(Base._return_type(*, eltype(w), valtype(ac)))
+    else
+        return sum(w[i] * star(v) for (i, v) in nonzero_pairs(ac))
+    end
 end
 
 function LinearAlgebra.dot(ac::AbstractCoefficients, w::AbstractVector)
     @assert key_type(ac) <: Integer
-    isempty(keys(ac)) && return zero(eltype(v))
-    return sum(v * w[i] for (i, v) in nonzero_pairs(ac))
+    if isempty(values(ac))
+        return zero(Base._return_type(*, eltype(w), valtype(ac)))
+    else
+        return sum(v * star(w[i]) for (i, v) in nonzero_pairs(ac))
+    end
 end
 
 # general mutable API
diff --git a/test/abstract_coeffs.jl b/test/abstract_coeffs.jl
index a1b9df9..c5b6536 100644
--- a/test/abstract_coeffs.jl
+++ b/test/abstract_coeffs.jl
@@ -100,4 +100,8 @@ end
 
     @test fP2m * fP3 == fP3 * fP2m == fPAlt
     @test iszero(fP2m * fP2)
+
+    @test norm(fP2m) == norm(P2m) == norm(fP2m)
+    v = coeffs(P2m, basis(fRG)) # an honest vector
+    @test dot(fP2m, fP2m) == dot(coeffs(fP2m), v) == dot(v, coeffs(fP2m))
 end
diff --git a/test/monoid_algebra.jl b/test/monoid_algebra.jl
index 094913e..92b22cc 100644
--- a/test/monoid_algebra.jl
+++ b/test/monoid_algebra.jl
@@ -46,9 +46,11 @@
             fX = AlgebraElement(coeffs(X, basis(fRG)), fRG)
             fY = AlgebraElement(coeffs(Y, basis(fRG)), fRG)
 
-            @test LinearAlgebra.dot(fX, coeffs(fX)) ≈
-                  norm(fX)^2 ≈
-                  LinearAlgebra.dot(coeffs(fX), fX)
+            @test dot(X, Y) == dot(fX, fY) == dot(coeffs(X), coeffs(Y))
+            @test dot(fX, coeffs(fY)) == dot(coeffs(fX), fY)
+
+            @test dot(X, X) ≈ norm(X)^2 ≈ dot(coeffs(X), coeffs(X))
+            @test dot(fX, fX) ≈ norm(fX)^2 ≈ dot(coeffs(fX), coeffs(fX))
 
             @test coeffs(fX) ==
                   coeffs(coeffs(fX, basis(RG)), basis(RG), basis(fRG))