## Ring

Theorem. In an abstract clone C, if 2 operations +,# ∈C2 commute (i.e. preserve each other):
x,y,z,t, (x+y)#(z+t) = (x#z)+(y#t)
and have identity elements, respectively 0,e
x,     x+0 = 0+x = x = x#e = e#x
Then +,# coincide and are commutative and associative.

Proof:
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)
Conversely, if + is commutative and associative then it commutes with itself:
(x+y)+(z+t) = x+(y+(z+t))=x+((y+z)+t)=x+((z+y)+t)= (x+z)+(y+t)

Definition of addition. In an abstract clone C, an addition is an operation + ∈ C2 ∩ Pol C that has an identity element 0∈ C0.

(To make sense of the claim that + has an identity element in any representation, 0 needs to belong to C0 ; 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 Cn an n-ary addition symbol ∑k<n xk , where ∑k<2 xk = x0+x1, and

n∈ℕ, = ∑(k<n+1) xk = xn + ∑k<n xk  = (∑k<n xk )+ xn

These formulas can be used backwards, with xn=0, to give ∑k<1 xk = x0 and ∑k<0 xk = 0.

Since Pol C is a clone and all additions with nonzero arity were defined from binary addition, they are also central (i.e. in  Pol C). The unary addition is 1∈Pol C.
However, 0 needs not be central. If Pol C meets C0 then C0 is a singleton, which implies that 0 is itself central; but nothing ensures that this happens, if multiplication is not commutative. We have only the following result:

Remark. If + is cancellative (in particular if it is a group) then 0 is central

Proof. ∀aC1, a+0 = a⋅1 = a⋅(1+0) = a+a⋅0 (thus ∀xax=ax+a⋅0)
If + is cancellative then a⋅0 = 0.∎

Theorem. In an abstract clone with an addition, any operation tCn for n>0 takes the form (∑k<n ak⋅πk,n ), i.e. (xk)k<n↦∑k<n akxk  for some family (ak)k<nC1n called the coordinates of t, which is unique if 0 is central.

Let us write the proof for n=2 (higher arities are similar): let #∈C2. Since + is central, + and # commute:
x#y = (x+0)#(0+y) = (x#0)+(0#y) = (1#0)⋅ x + (0#1)⋅ y
Thus, x#y = a0x + a1y where (a0,a1) is defined by a0= 1#0 and a1= 0#1.

To check the uniqueness, starting from arbitrary (a0,a1), let # defined by x#y = a0x + a1y .
Then, 1#0= a0⋅1 + a1⋅0
If 0 is central then 1#0=a0 , and similarly a1= 0#1.∎

Corollary. If C has an addition and C1 is commutative then all C is commutative (Pol C = C).

More generally, the above theorem shows that the whole structure of an abstract clone with an addition, is determined for all Cn with n>0 by the algebraic structure of its mere set C1 of scalars, with language (0,1,+,⋅); as for C0, it is reduced to {0} when 0 is central (otherwise it is also determined by C1 as in any abstract clone). This language describing C1 as an algebra, should not be confused with the language operating in representations, that is the whole set C of symbols (where multiplication is seen as a set of unary operations).

• (C1,⋅,1) forms a monoid, as in any abstract clone,
• (C1,+,0) forms a commutative monoid, a condition equivalent to (+ ∈ C2 ∩ 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 + ∈ Pol C1
• xC1, 0⋅x = 0, as in any abstract clone
• xC1, x⋅0 = 0 means 0 ∈ Pol C1, i.e. 0 is central
Now, from this to the full theory of rings and their modules (that were called "representations" for abstract clones), the only thing missing is to make + a group (by which 0 will be 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 C1), so that the full language of rings is in fact (0,1,-1,+,⋅)
We just need one axiom for it :
-1 + 1 = 0
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
Thus + is cancellative, thus 0 is central. Then, -1 is also central because:
Switching sides in the last line of proof gives (x⋅ -1) + x = 0 = -x + x
Thus, since + is cancellative, x⋅ -1= -x.
From the properties of groups comes -(-1) = 1.
Another important operation in rings and their modules is subtraction, which belongs to C2∩Pol C, with coordinates (1,-1).

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