Rings

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 C_n an n-ary addition symbol ∑_k<n x_k , where ∑_k<2 x_k = x₀+x₁, and

∀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 ∑_k<1 x_k = x₀ and ∑_k<0 x_k = 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. ∀a∈C₁, a+0 = a⋅1 = a⋅(1+0) = a+a⋅0 (thus ∀x, a⋅x=a⋅x+a⋅0)
If + is cancellative then a⋅0 = 0.∎

Theorem. In an abstract clone with an addition, any operation t ∈ C_n for n>0 takes the form (∑_k<n a_k⋅π_k,n ), i.e. (x_k)_k<n↦∑_k<n a_k⋅x_k for some family (a_k)_k<n∈C₁ⁿ 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 = a₀⋅ x + a₁⋅ y 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 = a₀⋅ x + a₁⋅ y .
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 C_n 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).

Let us repeat the axioms already seen about C₁:

(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
∀x∈C₁, 0⋅x = 0, as in any abstract clone
∀x∈C₁, 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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