diff --git a/Project.toml b/Project.toml
index c15a66b789..326f904601 100644
--- a/Project.toml
+++ b/Project.toml
@@ -28,7 +28,7 @@ GAPExt = "GAP"
 PolymakeExt = "Polymake"
 
 [compat]
-AbstractAlgebra = "^0.43.1"
+AbstractAlgebra = "^0.43.10"
 Dates = "1.6"
 Distributed = "1.6"
 GAP = "0.9.6, 0.10, 0.11, 0.12"
diff --git a/src/Sparse/Row.jl b/src/Sparse/Row.jl
index 96c4eb82d0..653ba87d2c 100644
--- a/src/Sparse/Row.jl
+++ b/src/Sparse/Row.jl
@@ -105,6 +105,14 @@ function Base.empty!(A::SRow)
   return A
 end
 
+function Base.empty(A::SRow)
+  return sparse_row(base_ring(A))
+end
+
+function zero(A::SRow)
+  return empty(A)
+end
+
 function swap!(A::SRow, B::SRow)
   A.pos, B.pos = B.pos, A.pos
   A.values, B.values = B.values, A.values
@@ -447,6 +455,7 @@ end
 #  Inplace scaling
 #
 ################################################################################
+
 @doc raw"""
     scale_row!(a::SRow, b::NCRingElem) -> SRow
 
@@ -454,8 +463,9 @@ Returns the (left) product of $b \times a$ and reassigns the value of $a$ to thi
 For rows, the standard multiplication is from the left.
 """
 function scale_row!(a::SRow{T}, b::T) where T
-  @assert !iszero(b)
-  if isone(b)
+  if iszero(b)
+    return empty!(a)
+  elseif isone(b)
     return a
   end
   i = 1
@@ -465,20 +475,23 @@ function scale_row!(a::SRow{T}, b::T) where T
       deleteat!(a.values, i)
       deleteat!(a.pos, i)
     else
-     i += 1
+      i += 1
     end
   end
   return a
 end
 
+scale_row!(a::SRow, b) = scale_row!(a, base_ring(a)(b))
+
 @doc raw"""
     scale_row_right!(a::SRow, b::NCRingElem) -> SRow
 
 Returns the (right) product of $a \times b$ and modifies $a$ to this product.
 """
 function scale_row_right!(a::SRow{T}, b::T) where T
-  @assert !iszero(b)
-  if isone(b)
+  if iszero(b)
+    return empty!(a)
+  elseif isone(b)
     return a
   end
   i = 1
@@ -488,16 +501,20 @@ function scale_row_right!(a::SRow{T}, b::T) where T
       deleteat!(a.values, i)
       deleteat!(a.pos, i)
     else
-     i += 1
+      i += 1
     end
   end
   return a
 end
 
+scale_row_right!(a::SRow, b) = scale_row_right!(a, base_ring(a)(b))
+
 function scale_row_left!(a::SRow{T}, b::T) where T
   return scale_row!(a,b)
 end
 
+scale_row_left!(a::SRow, b) = scale_row_left!(a, base_ring(a)(b))
+
 ################################################################################
 #
 #  Addition
@@ -506,9 +523,9 @@ end
 
 function +(A::SRow{T}, B::SRow{T}) where T
   if length(A.values) == 0
-    return B
+    return deepcopy(B)
   elseif length(B.values) == 0
-    return A
+    return deepcopy(A)
   end
   return add_scaled_row(A, B, one(base_ring(A)))
 end
@@ -516,12 +533,12 @@ end
 function -(A::SRow{T}, B::SRow{T}) where T
   if length(A) == 0
     if length(B) == 0
-      return A
+      return deepcopy(A)
     else
-      return add_scaled_row(B, A, base_ring(B)(-1))
+      return add_scaled_row(B, A, -1)
     end
   end
-  return add_scaled_row(B, A, base_ring(A)(-1))
+  return add_scaled_row(B, A, -1)
 end
 
 function -(A::SRow{T}) where {T}
@@ -683,10 +700,10 @@ end
 
 Returns the row $c A + B$.
 """
-add_scaled_row(a::SRow{T}, b::SRow{T}, c::T) where {T} = add_scaled_row!(a, deepcopy(b), c)
+add_scaled_row(a::SRow{T}, b::SRow{T}, c) where {T} = add_scaled_row!(a, deepcopy(b), c)
 
-add_left_scaled_row(a::SRow{T}, b::SRow{T}, c::T) where {T} = add_left_scaled_row!(a, deepcopy(b), c)
-add_right_scaled_row(a::SRow{T}, b::SRow{T}, c::T) where {T} = add_right_scaled_row!(a, deepcopy(b), c)
+add_left_scaled_row(a::SRow{T}, b::SRow{T}, c) where {T} = add_left_scaled_row!(a, deepcopy(b), c)
+add_right_scaled_row(a::SRow{T}, b::SRow{T}, c) where {T} = add_right_scaled_row!(a, deepcopy(b), c)
 
 
 
@@ -696,7 +713,9 @@ add_right_scaled_row(a::SRow{T}, b::SRow{T}, c::T) where {T} = add_right_scaled_
 Adds the left scaled row $c A$ to $B$.
 """
 function add_scaled_row!(a::SRow{T}, b::SRow{T}, c::T, ::Val{left_side} = Val(true)) where {T, left_side}
-  @assert a !== b
+  if a === b
+    a = deepcopy(a)
+  end
   i = 1
   j = 1
   t = base_ring(a)()
@@ -735,17 +754,144 @@ function add_scaled_row!(a::SRow{T}, b::SRow{T}, c::T, ::Val{left_side} = Val(tr
   return b
 end
 
+add_scaled_row!(a::SRow{T}, b::SRow{T}, c) where {T} = add_scaled_row!(a, b, base_ring(a)(c))
+
+add_scaled_row!(a::SRow{T}, b::SRow{T}, c, side::Val) where {T} = add_scaled_row!(a, b, base_ring(a)(c), side)
+
 # ignore tmp argument
-add_scaled_row!(a::SRow{T}, b::SRow{T}, c::T, tmp::SRow{T}) where T = add_scaled_row!(a, b, c)
+add_scaled_row!(a::SRow{T}, b::SRow{T}, c, tmp::SRow{T}) where T = add_scaled_row!(a, b, c)
 
-add_left_scaled_row!(a::SRow{T}, b::SRow{T}, c::T) where T = add_scaled_row!(a, b, c)
+add_left_scaled_row!(a::SRow{T}, b::SRow{T}, c) where T = add_scaled_row!(a, b, c)
 
 @doc raw"""
     add_right_scaled_row!(A::SRow{T}, B::SRow{T}, c::T) -> SRow{T}
 
-Return the right scaled row $c A$ to $B$ by changing $B$ in place.
+Return the right scaled row $A c$ to $B$ by changing $B$ in place.
 """
-add_right_scaled_row!(a::SRow{T}, b::SRow{T}, c::T) where T = add_scaled_row!(a, b, c, Val(false))
+add_right_scaled_row!(a::SRow{T}, b::SRow{T}, c) where T = add_scaled_row!(a, b, c, Val(false))
+
+
+################################################################################
+#
+#  Mutating arithmetics
+#
+################################################################################
+
+function zero!(z::SRow)
+  return empty!(z)
+end
+
+function neg!(z::SRow{T}, x::SRow{T}) where T
+  if z === x
+    return neg!(x)
+  end
+  swap!(z, -x)
+  return z
+end
+
+function neg!(z::SRow)
+  for i in 1:length(z)
+    z.values[i] = neg!(z.values[i])
+  end
+  return z
+end
+
+function add!(z::SRow{T}, x::SRow{T}, y::SRow{T}) where T
+  if z === x
+    return add!(x, y)
+  elseif z === y
+    return add!(y, x)
+  end
+  swap!(z, x + y)
+  return z
+end
+
+function add!(z::SRow{T}, x::SRow{T}) where T
+  if z === x
+    return scale_row!(z, 2)
+  end
+  return add_scaled_row!(x, z, one(base_ring(x)))
+end
+
+function sub!(z::SRow{T}, x::SRow{T}, y::SRow{T}) where T
+  if z === x
+    return sub!(x, y)
+  elseif z === y
+    return neg!(sub!(y, x))
+  end
+  swap!(z, x - y)
+  return z
+end
+
+function sub!(z::SRow{T}, x::SRow{T}) where T
+  if z === x
+    return empty!(z)
+  end
+  return add_scaled_row!(x, z, -1)
+end
+
+function mul!(z::SRow{T}, x::SRow{T}, y::SRow{T}) where T
+  error("Not implemented")
+end
+
+function mul!(z::SRow{T}, x::SRow{T}, c) where T
+  if z === x
+    return scale_row_right!(x, c)
+  end
+  swap!(z, x * c)
+  return z
+end
+
+function mul!(z::SRow{T}, c, y::SRow{T}) where T
+  if z === y
+    return scale_row_left!(y, c)
+  end
+  swap!(z, c * y)
+  return z
+end
+
+function addmul!(z::SRow{T}, x::SRow{T}, y::SRow{T}) where T
+  error("Not implemented")
+end
+
+function addmul!(z::SRow{T}, x::SRow{T}, y) where T
+  if z === x
+    return scale_row_right!(x, y+1)
+  end
+  return add_right_scaled_row!(x, z, y)
+end
+
+function addmul!(z::SRow{T}, x, y::SRow{T}) where T
+  if z === x
+    return scale_row_left!(y, x+1)
+  end
+  return add_left_scaled_row!(y, z, x)
+end
+
+function submul!(z::SRow{T}, x::SRow{T}, y::SRow{T}) where T
+  error("Not implemented")
+end
+
+function submul!(z::SRow{T}, x::SRow{T}, y) where T
+  if z === x
+    return scale_row_right!(x, -y+1)
+  end
+  return add_right_scaled_row!(x, z, -y)
+end
+
+function submul!(z::SRow{T}, x, y::SRow{T}) where T
+  if z === x
+    return scale_row_left!(y, -x+1)
+  end
+  return add_left_scaled_row!(y, z, -x)
+end
+
+
+# ignore temp variable
+addmul!(z::SRow{T}, x::SRow{T}, y, t) where T = addmul!(z, x, y)
+addmul!(z::SRow{T}, x, y::SRow{T}, t) where T = addmul!(z, x, y)
+submul!(z::SRow{T}, x::SRow{T}, y, t) where T = submul!(z, x, y)
+submul!(z::SRow{T}, x, y::SRow{T}, t) where T = submul!(z, x, y)
 
 
 ################################################################################
diff --git a/src/Sparse/ZZRow.jl b/src/Sparse/ZZRow.jl
index d4cfc30f04..a9073f5a23 100644
--- a/src/Sparse/ZZRow.jl
+++ b/src/Sparse/ZZRow.jl
@@ -276,7 +276,6 @@ end
 
 function add_scaled_row(Ai::SRow{ZZRingElem}, Aj::SRow{ZZRingElem}, c::ZZRingElem, sr::SRow{ZZRingElem} = sparse_row(ZZ))
   empty!(sr)
-  @assert c != 0
   n = ZZRingElem()
   pi = 1
   pj = 1
@@ -323,6 +322,9 @@ function add_scaled_row(Ai::SRow{ZZRingElem}, Aj::SRow{ZZRingElem}, c::ZZRingEle
 end
 
 function add_scaled_row!(Ai::SRow{ZZRingElem}, Aj::SRow{ZZRingElem}, c::ZZRingElem, sr::SRow{ZZRingElem} = sparse_row(ZZ))
+  if iszero(c)
+    return Aj
+  end
   _t = sr
   sr = add_scaled_row(Ai, Aj, c, sr)
   @assert _t === sr
diff --git a/test/Sparse/Row.jl b/test/Sparse/Row.jl
index d104b80cca..14c3fb12bf 100644
--- a/test/Sparse/Row.jl
+++ b/test/Sparse/Row.jl
@@ -204,4 +204,40 @@
   B = sparse_row(F,[1],[y])
   C = add_scaled_row(A,B,F(1))
   @test C == A+B
+
+  # mutating arithmetic
+  randcoeff() = begin
+    n = rand((1,1,1,2,5,7,15))
+    return rand(-2^n:2^n)
+  end
+  Main.equality(A::SRow, B::SRow) = A == B
+  @testset "mutating arithmetic; R = $R" for R in (ZZ, QQ)
+    for _ in 1:10
+      maxind_A = rand(0:10)
+      inds_A = Hecke.Random.randsubseq(1:maxind_A, rand())
+      vals_A = elem_type(R)[R(rand((-1, 1)) * rand(1:10)) for _ in 1:length(inds_A)]
+      A = sparse_row(R, inds_A, vals_A)
+
+      maxind_B = rand(0:10)
+      inds_B = Hecke.Random.randsubseq(1:maxind_B, rand())
+      vals_B = elem_type(R)[R(rand((-1, 1)) * rand(1:10)) for _ in 1:length(inds_B)]
+      B = sparse_row(R, inds_B, vals_B)
+
+      test_mutating_op_like_zero(zero, zero!, A)
+
+      test_mutating_op_like_neg(-, neg!, A)
+
+      test_mutating_op_like_add(+, add!, A, B)
+      test_mutating_op_like_add(-, sub!, A, B)
+      test_mutating_op_like_add(*, mul!, A, randcoeff(), SRow)
+      test_mutating_op_like_add(*, mul!, randcoeff(), A, SRow)
+      test_mutating_op_like_add(*, mul!, A, ZZ(randcoeff()), SRow)
+      test_mutating_op_like_add(*, mul!, ZZ(randcoeff()), A, SRow)
+
+      test_mutating_op_like_addmul((a, b, c) -> a + b*c, addmul!, A, B, randcoeff(), SRow)
+      test_mutating_op_like_addmul((a, b, c) -> a + b*c, addmul!, A, randcoeff(), B, SRow)
+      test_mutating_op_like_addmul((a, b, c) -> a - b*c, submul!, A, B, randcoeff(), SRow)
+      test_mutating_op_like_addmul((a, b, c) -> a - b*c, submul!, A, randcoeff(), B, SRow)
+    end
+  end
 end