definition of integers modulo natural number ring
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of integers modulo natural number group.
- The reader knows a definition of ring.
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:
\(*\mathbb{Z} / n\): \(= \text{ the integers modulo natural number group }\) with the multiplication specified below, \(\in \{\text{ the commutative rings }\}\)
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Conditions:
\(\forall [z_1], [z_2] \in \mathbb{Z} / n ([z_1] [z_2] = [z_1 z_2])\)
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2: Note
Let us see that the multiplication is well-defined.
That is about that \([z_1 z_2]\) does not depend on the representatives, \(z_1, z_2\).
Let \(z'_1, z'_2 \in \mathbb{Z}\) be any such that \([z_1] = [z'_1]\) and \([z_2] = [z'_2]\). That means that \(z'_1 = z_1 + l_1 n\) and \(z'_2 = z_2 + l_2 n\). \([z'_1 z'_2] = [(z_1 + l_1 n) (z_2 + l_2 n)] = [z_1 z_2 + n (z_1 l_2 + l_1 z_2 + l_1 l_2 n)] = [z_1 z_2]\).
Let us see that \(\mathbb{Z} / n\) is indeed a commutative ring.
The operation is closed, because \([z_1 z_2] \in \mathbb{Z} / n\).
\([1]\) is the identity element: \([1] [z] = [1 z] = [z]\) and \([z] [1] = [z 1] = [z]\).
The multiplication is associative: for each \([z_1], [z_2], [z_3] \in \mathbb{Z} / n\), \(([z_1] [z_2]) [z_3] = [z_1 z_2] [z_3] = [z_1 z_2 z_3] = [z_1] [z_2 z_3] = [z_1] ([z_2] [z_3])\).
The addition and the multiplication are distributive: for each \([z_1], [z_2], [z_3] \in \mathbb{Z} / n\), \([z_1] ([z_2] + [z_3]) = [z_1] [z_2 + z_3] = [z_1 (z_2 + z_3)] = [z_1 z_2 + z_1 z_3] = [z_1 z_2] + [z_1 z_3] = [z_1] [z_2] + [z_1] [z_3]\); \(([z_1] + [z_2]) [z_3] = [z_1 + z_2] [z_3] = [(z_1 + z_2) z_3] = [z_1 z_3 + z_2 z_3] = [z_1 z_3] + [z_2 z_3] = [z_1] [z_3] + [z_2] [z_3]\).
So, \(\mathbb{Z} / n\) is a ring.
\(\mathbb{Z} / n\) is a commutative ring: for each \([z_1], [z_2] \in \mathbb{R} / n\), \([z_1] [z_2] = [z_1 z_2] = [z_2 z_1] = [z_2] [z_1]\).
In fact, \(\mathbb{Z} / n = \mathbb{Z} / (n \mathbb{Z})\), the quotient ring by the principal ideal, because the multiplication for the quotient ring is exactly the multiplication specified in this definition.
