diff --git a/src/arithmetic.jl b/src/arithmetic.jl
index 9645323..346a1a4 100644
--- a/src/arithmetic.jl
+++ b/src/arithmetic.jl
@@ -30,12 +30,24 @@ end
 function MA.promote_operation(::typeof(*), ::Type{T}, ::Type{A}) where {T<:Number, A<:AlgebraElement}
     return algebra_promote_operation(*, A, T)
 end
-function Base.:*(a::Number, X::AlgebraElement)
-    return MA.operate_to!(_preallocate_output(*, X, a), *, X, a)
+function Base.:*(a::Any, X::AlgebraElement)
+    return MA.operate_to!(_preallocate_output(*, X, a), *, a, X)
 end
 function Base.:div(X::AlgebraElement, a::Number)
     return MA.operate_to!(_preallocate_output(div, X, a), div, X, a)
 end
+function Base.:*(
+    a::T,
+    X::AlgebraElement{A},
+) where {T,O,A<:AbstractStarAlgebra{O,T}}
+    return MA.operate_to!(similar(X), __lmul, a, X)
+end
+function Base.:*(
+    X::AlgebraElement{A},
+    a::T,
+) where {T,O,A<:AbstractStarAlgebra{O,T}}
+    return MA.operate_to!(similar(X), __rmul, a, X)
+end
 
 for op in [:+, :-, :*]
     @eval begin
@@ -64,11 +76,11 @@ end
 function MA.operate_to!(
     res::AlgebraElement,
     ::typeof(*),
+    a::Any,
     X::AlgebraElement,
-    a::Number,
 )
     @assert parent(res) === parent(X)
-    MA.operate_to!(coeffs(res), *, coeffs(X), a)
+    MA.operate_to!(coeffs(res), *, a, coeffs(X))
     return res
 end
 
@@ -83,6 +95,17 @@ function MA.operate_to!(
     return res
 end
 
+function MA.operate_to!(
+    res::AlgebraElement,
+    mul::Union{typeof(__lmul),typeof(__rmul)},
+    a,
+    X::AlgebraElement,
+)
+    @assert parent(res) == parent(X)
+    MA.operate_to!(coeffs(res), mul, a, coeffs(X))
+    return res
+end
+
 function MA.operate_to!(res::AlgebraElement, ::typeof(-), X::AlgebraElement)
     @assert parent(res) === parent(X)
     MA.operate_to!(coeffs(res), -, coeffs(X))
@@ -118,9 +141,7 @@ function MA.operate_to!(
     args::Vararg{AlgebraElement,N},
 ) where {N}
     for arg in args
-        if arg isa AlgebraElement
-            @assert parent(res) == parent(arg)
-        end
+        @assert parent(res) == parent(arg)
     end
     mstr = mstructure(basis(res))
     MA.operate_to!(coeffs(res), mstr, coeffs.(args)...)
@@ -133,9 +154,7 @@ function MA.operate!(
     args::Vararg{AlgebraElement,N},
 ) where {N}
     for arg in args
-        if arg isa AlgebraElement
-            @assert parent(res) == parent(arg)
-        end
+        @assert parent(res) == parent(arg)
     end
     mstr = mstructure(basis(res))
     MA.operate!(UnsafeAddMul(mstr), coeffs(res), coeffs.(args)...)
diff --git a/src/coefficients.jl b/src/coefficients.jl
index d6edde2..9330350 100644
--- a/src/coefficients.jl
+++ b/src/coefficients.jl
@@ -122,12 +122,12 @@ end
 
 Base.zero(X::AbstractCoefficients) = MA.operate!(zero, similar(X))
 Base.:-(X::AbstractCoefficients) = MA.operate_to!(__prealloc(X, -1, *), -, X)
-Base.:*(a::Number, X::AbstractCoefficients) = X * a
-Base.:/(X::AbstractCoefficients, a::Number) = X * inv(a)
+Base.:*(X::AbstractCoefficients, a::Any) = a * X
+Base.:/(X::AbstractCoefficients, a::Number) = inv(a) * X
 Base.://(X::AbstractCoefficients, a::Number) = X * 1 // a
 
-function Base.:*(X::AbstractCoefficients, a::Number)
-    return MA.operate_to!(__prealloc(X, a, *), *, X, a)
+function Base.:*(a::Any, X::AbstractCoefficients)
+    return MA.operate_to!(__prealloc(X, a, *), *, a, X)
 end
 function Base.:div(X::AbstractCoefficients, a::Number)
     return MA.operate_to!(__prealloc(X, a, div), div, X, a)
@@ -167,8 +167,8 @@ end
 function MA.operate_to!(
     res::AbstractCoefficients,
     ::typeof(*),
+    a::Any,
     X::AbstractCoefficients,
-    a::Number,
 )
     if res !== X
         MA.operate!(zero, res)
@@ -240,3 +240,25 @@ function MA.operate_to!(
     end
     return res
 end
+
+__lmul(a, k) = a * k
+__rmul(a, k) = k * a
+
+function MA.operate_to!(
+    res::AbstractCoefficients,
+    mul::Union{typeof(__lmul),typeof(__rmul)},
+    a,
+    X::AbstractCoefficients,
+)
+    if res === X
+        throw("No aliasing is allowed in shift")
+    else
+        MA.operate!(zero, res)
+        for (k, v) in nonzero_pairs(X)
+            res[mul(a, k)] += v
+        end
+    end
+    return res
+end
+
+
diff --git a/test/abstract_coeffs.jl b/test/abstract_coeffs.jl
index 4474945..c4aa1d5 100644
--- a/test/abstract_coeffs.jl
+++ b/test/abstract_coeffs.jl
@@ -60,4 +60,10 @@
     @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))
+
+    s1, s2 = PermutationGroups.gens(G)
+    @assert s1 * s2 ≠ s2 * s1
+    Z = RG(1) + RG(s1)
+    @test s2 * Z == RG(s2) + RG(s2 * s1)
+    @test Z * s2 == RG(s2) + RG(s1 * s2)
 end
diff --git a/test/monoid_algebra.jl b/test/monoid_algebra.jl
index 9a3117c..8be47ca 100644
--- a/test/monoid_algebra.jl
+++ b/test/monoid_algebra.jl
@@ -73,8 +73,8 @@
             @test coeffs(Z) == coeffs(fZ, basis(RG))
             @test coeffs(fZ) == coeffs(Z, basis(fRG))
 
-            @test coeffs(2 * X * Y) == coeffs(MA.operate_to!(Z, *, Z, 2))
-            @test coeffs(2 * fX * fY) == coeffs(MA.operate_to!(fZ, *, fZ, 2))
+            @test coeffs(2 * X * Y) == coeffs(MA.operate_to!(Z, *, 2, Z))
+            @test coeffs(2 * fX * fY) == coeffs(MA.operate_to!(fZ, *, 2, fZ))
         end
     end
     @testset "mutable arithmetic" begin
@@ -118,19 +118,19 @@
         end
 
         let d = deepcopy(a)
-            MA.operate_to!(d, *, a, 2)
+            MA.operate_to!(d, *, 2, a)
             @test d == 2a
-            MA.operate_to!(d, *, d, 2)
+            MA.operate_to!(d, *, 2, d)
             @test d == 4a
             MA.operate_to!(d, *, a, b)
             @test d == a * b
             @test_throws ArgumentError MA.operate_to!(d, *, d, b)
 
             d = deepcopy(a)
-            @test @allocated(MA.operate_to!(d, *, d, 2)) == 0
+            @test @allocated(MA.operate_to!(d, *, 2, d)) == 0
             @test d == 2a
 
-            @test @allocated(MA.operate_to!(d, *, a, 2)) == 0
+            @test @allocated(MA.operate_to!(d, *, 2, a)) == 0
             @test d == 2a
 
             MA.operate!(zero, d)