∀*x*,*y*,*z*,*t*, (*x*+*y*)#(*z*+*t*)
= (*x*#*z*)+(*y*#*t*)

and have identity elements, respectively 0,∀*x*, *x*+0 = 0+*x*
= *x* = *x*#*e* = *e*#*x*

Then +,# coincide and are commutative and associative.Proof:

Conversely, if + is commutative and associative then it commutes with itself:e=e#e= (0+e)#(e+0) = (0#e)+(e#0) = 0+0 = 0

∀x,y,x+y= (x#e)+(e#y) = (x+0)#(0+y) =x#y= (0+x)#(y+0) = (e#y)+(x#e) =y+x

∀x,y,z,(x+y)+z=(x+y)+(0+z) = (x+0)+(y+z) =x+(y+z)

(

(To make sense of the claim that + has an identity element in any
representation, 0 needs to belong to *C*_{0} ; we
need to add there, otherwise, morphisms may not preserve it, but
then representations where + has an identity, would not form a
variety anymore)

The above theorem ensures that, in any clone, there can only be
one addition, and it is commutative and associative;
multiplication is distributive over addition on both sides.

For any *n*∈ℕ we also have in *C _{n}* an

∀*n*∈ℕ, = ∑_{(k<}_{n+1)
}*x*_{k} = *x*_{n} + ∑_{k<n}
*x*_{k} = (∑_{k<n} *x*_{k}
)+ *x*_{n}

These formulas can be used backwards, with *x _{n}*=0,
to give ∑

However, 0 needs not be central. If Pol

Proof. ∀

If + is cancellative then

**Theorem.** In an abstract clone with an addition, any
operation *t* ∈ *C _{n}* for

To check the uniqueness, starting from arbitrary (

Then, 1#0=

If 0 is central then 1#0=

More generally, the above theorem shows that the whole structure of an abstract clone with an addition, is determined for all

Let us repeat the axioms already seen about

- (
*C*_{1},⋅,1) forms a monoid, as in any abstract clone, - (
*C*_{1},+,0) forms a commutative monoid, a condition equivalent to (+ ∈*C*_{2}∩ Pol{+} and 0 is an identity of +) - ⋅ is right distributive over +, as over any operation in an abstract clone;
- ⋅ is left distributive over +, which means + ∈
*C*_{1} - ∀
*x*∈*C*_{1}, 0⋅*x*= 0, as in any abstract clone - ∀
*x*∈*C*_{1},*x*⋅0 = 0 means 0 ∈*C*_{1}, i.e. 0 is central

As its inversion function in any module is a unary operation, it is a scalar, named -1, to be added as a constant in the language of rings (in

We just need one axiom for it :

-1 + 1 = 0

Thus + is cancellative, thus 0 is central. Then, -1 is also central
because:Indeed, it gives an inverse for addition to any
*x* of any representation, that is (-1⋅*x*), denoted -*x*
(coherent with -1⋅1= -1).

∀*x*, -*x* + *x* = (-1⋅*x*) + 1⋅*x*
= (-1+ 1)⋅*x* = 0⋅*x* = 0

∀

Switching sides in the last line of proof gives (From the properties of groups comes -(-1) = 1.x⋅ -1) +x= 0 = -x+x

Thus, since + is cancellative,x⋅ -1= -x.

Another important operation in rings and their modules is

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