2024-12-01

875: Integers Modulo Natural Number Ring

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definition of integers modulo natural number ring

Topics


About: ring

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of integers modulo natural number ring.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

Main Body

1: Structured Description


Here is the rules of Structured Description.

Entities:
∗Z/n: = the integers modulo natural number group  with the multiplication specified below, ∈{ the commutative rings }
//

Conditions:
∀[z1],[z2]∈Z/n([z1][z2]=[z1z2])
//

2: Note


Let us see that the multiplication is well-defined.

That is about that [z1z2] does not depend on the representatives, z1,z2.

Let z1′,z2′∈Z be any such that [z1]=[z1′] and [z2]=[z2′]. That means that z1′=z1+l1n and z2′=z2+l2n. [z1′z2′]=[(z1+l1n)(z2+l2n)]=[z1z2+n(z1l2+l1z2+l1l2n)]=[z1z2].

Let us see that Z/n is indeed a commutative ring.

The operation is closed, because [z1z2]∈Z/n.

[1] is the identity element: [1][z]=[1z]=[z] and [z][1]=[z1]=[z].

The multiplication is associative: for each [z1],[z2],[z3]∈Z/n, ([z1][z2])[z3]=[z1z2][z3]=[z1z2z3]=[z1][z2z3]=[z1]([z2][z3]).

The addition and the multiplication are distributive: for each [z1],[z2],[z3]∈Z/n, [z1]([z2]+[z3])=[z1][z2+z3]=[z1(z2+z3)]=[z1z2+z1z3]=[z1z2]+[z1z3]=[z1][z2]+[z1][z3]; ([z1]+[z2])[z3]=[z1+z2][z3]=[(z1+z2)z3]=[z1z3+z2z3]=[z1z3]+[z2z3]=[z1][z3]+[z2][z3].

So, Z/n is a ring.

Z/n is a commutative ring: for each [z1],[z2]∈R/n, [z1][z2]=[z1z2]=[z2z1]=[z2][z1].

In fact, Z/n=Z/(nZ), the quotient ring by the principal ideal, because the multiplication for the quotient ring is exactly the multiplication specified in this definition.

References
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