Construction of the relative entireties

The goal of this article is of:

• to build $\backslash \; mathbb\; \{Z\}$ like a unit,
• to build a structure of group on this unit,
• to show that prolongs the monoid (additive) natural whole ,

The structure of ring will be just outlined.

Construction of the unit Z

It is known already that the whole of the natural whole $\backslash \; mathbb\; \{NR\}$, provided with the operation interns addition, is a commutative Monoïde; thus our goal is simply to add an opposite (opposite for the addition) for each entirety not no one. It is not a question to brutally add an element, it is also necessary to give the means of defining the addition without pain!

This is why one will leave the naive concept of relative entirety, which one supposes already known, to build the mathematical object corresponding. If one wants to define $-2$ with natural entireties, one wants to see it like $0-2$, or $5-7$, or…; in short, one wants to see it like the difference in two natural entireties. That raises a difficulty, because it is seen on the one hand that the writing is not single, and on the other hand, that utilized an operation, the subtraction, which does not have any direction with the natural entireties!

One thus will consider pairs of entireties, form $(n\_1,\; n\_2)$, and will consider that the pair $(n\_1,\; n\_2)$ corresponds to the naive relative entirety $n\_1-n\_2$; and as it was seen that it is not reasonable to take $\backslash \; mathbb\; \{NR\}\; \backslash \; times\; \backslash \; mathbb\; \{NR\}$ like whole of the relative entireties, it will gather the pairs which correspond to the same naive relative entirety.

For that, one will define on $\backslash \; mathbb\; \{NR\}\; \backslash \; times\; \backslash \; mathbb\; \{NR\}$ a Relation of equivalence $R$, by the following relation: $(n\_1,\; n\_2)\; R\; (n\_1\text{'},\; n\_2\text{'})\; \backslash \; Leftrightarrow\; n\_1+n\_2\text{'}=n\_1\text{'}+n\_2$. Note how intuitively one is writing that two couples are equal so when one withdraws the second of the pair from the first one obtains the same relative entirety! But one uses only the sum to define $R$, therefore this definition does not use a naive object.

The relations of equivalences are made for quotienter; one thus defines: $\backslash \; mathbb\; \{Z\}\; =\; (\backslash \; mathbb\; \{NR\}\; \backslash \; times\; \backslash \; mathbb\; \{NR\})\; /R$

Definition of the structure of group

One has the whole of the relative entireties now; it remains to define the addition on the latter: for that, one has only the definition on the entireties; one thus initially will define an operation on the pairs of entireties, and as it will be compatible with the relation $R$, it will give an operation on the relative entireties!

One as follows defines the sum of two pairs of entirety: $(n\_1,\; n\_2)\; +\; (n\_1\text{'},\; n\_2\text{'})\; =\; (n\_1+n\_1\text{'},\; n\_2+n\_2\text{'})$; this operation is obviously already commutative, associative and of neutral element $(0,0)$ on the pairs of entireties; it passes clearly to the quotient, to give on $\backslash \; mathbb\; \{Z\}$ a structure of commutative monoid.

It thus remains only to find an opposite with entire relative; but this is immediate: if $(n\_1,\; n\_2)$ represents a relative entirety in the pairs of entireties, there is $(n\_1,\; n\_2)\; +\; (n\_2,\; n\_1)\; =\; (n\_1+n\_2,\; n\_1+n\_2)$ where $(n\_1+n\_2,\; n\_1+n\_2)$ is equivalent to $(0,0)$, therefore the class of equivalence of $(n\_2,\; n\_1)$ is opposed to the class of equivalence of $(n\_1,\; n\_2)$…

Checking of the prolongation

One will show that there is a Morphisme monoids injective of $\backslash \; mathbb\; \{NR\}$ in $\backslash \; mathbb\; \{Z\}$; in this way, one will be able to see a natural entirety like a particular case of relative entirety. Again, it is the naive idea which one had of the relative entireties which shows the way.

That is to say $n$ a natural entirety; one associates to him the class of the pair $(N,\; 0)$. It is seen whereas:

• $0$ has as an image the class of $(0,0)$, therefore the $0$ of the relative entireties;
• $n+n\text{'}$, the sum of two entireties, has as an image the class of $(n+n\text{'},\; 0)$, which is the sum of the classes of $(N,\; 0)$ and $(,\; 0)$.

in addition, it is seen well that this application is injective, since to require that the classes of $(N,\; 0)$ and $(,\; 0)$ be equal, it is precisely to require that $n=n\text{'}$!

Simplified writing of the elements of Z

Any couple of natural entireties ( N; m ) is in one of these three types of classes

• a class ( D; 0 ) if N > m with N = m + D and D not no one
• a class ( 0; D ) if N < m with N + D = m and D not no one
• the class (0; 0) if N = m

However the whole of the classes ( D; 0 ) is isomorphous with $\backslash \; mathbb\; \{NR\}$, one thus notes these classes in the simplified form D .

In addition, for D not no one, the classes ( D; 0 ) and ( 0; D ) are opposite. Indeed, ( D; 0 ) + ( 0; D ) = ( D ; D ) = (0; 0) in term of classes. One thus notes the classes ( 0; D ) in the simplified form (- D ).

The unit $\backslash \; mathbb\; \{Z\}$ then finds its more traditional form of $\backslash \; mathbb\; \{NR\}\; \backslash \; cup\; \backslash \; \{(-\; D)\; /d\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; ^*\; \backslash \}$.

Definition of the multiplication

One can then define the multiplication as follows: $(n\_1,\; n\_2)\; \backslash \; times\; (m\_1,\; m\_2)\; =\; (n\_1\; m\_1\; +\; n\_2\; m\_2,\; n\_1\; m\_2\; +\; m\_1\; n\_2)$ (always while taking as a starting point the analogy with the naive relative entireties).

This operation defined on $\backslash \; mathbb\; \{NR\}\; \backslash \; times\; \backslash \; mathbb\; \{NR\}$ is associative, commutative, has a neutral element (1; 0) and are distributive for the addition previously established. Moreover it is compatible with the relation of equivalence. By passage to the quotient, it confers on $\backslash \; mathbb\; \{Z\}$ a structure of unit ring.

The equalities

$(D;\; 0)\; \backslash \; times\; (of;\; 0)\; =\; (dd\text{'};\; 0)$

$(D;\; 0)\; \backslash \; times\; (0;\; of)\; =\; (0;\; dd\text{'})$

$(0;\; d)\; \backslash \; times\; (0;\; of)\; =\; (dd\text{'};\; 0)$

the writings

$d\; \backslash \; times\; D\; allow\; =\; dd\text{'}$

$d\; \backslash \; times\; (-\; of)\; =\; (-\; dd\text{'})$

$(-\; d)\; \backslash \; times\; (-\; of)\; =\; dd\text{'}$

This writing makes it possible to prove that the ring is also just.

Random links:

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