description/proof of that characteristic of integral domain is 0 or prime number
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Note 1
- 3: Proof
- 4: Note 2
Starting Context
- The reader knows a definition of characteristic of ring.
Target Context
- The reader will have a description and a proof of the proposition that the characteristic of any integral domain is 0 or a prime number.
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:
\(R\): \(\in \{\text{ the integral domains }\}\)
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Statements:
\(Ch (R) = 0 \lor Ch (R) \in \{\text{ the prime numbers }\}\)
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2: Note 1
Any field is an integral domain, and so, the characteristic of any field is \(0\) or a prime number.
3: Proof
Whole Strategy: Step 1: suppose that \(Ch (R) \neq 0\); Step 2: suppose that \(Ch (R) = m n\) where \(m, n \in \mathbb{N} \setminus \{0, 1\}\), and find a contradiction.
Step 1:
Let us suppose that \(Ch (R) \neq 0\).
Step 2:
Let us suppose that \(Ch (R)\) was not any prime number.
That would mean that \(Ch (R) = m n\) where \(m, n \in \mathbb{N} \setminus \{0, 1\}\).
\((m n) \cdot 1 = (m \cdot 1) (n \cdot 1) = 0\), by the distribution law.
\(m \cdot 1 = 0\) or \(n \cdot 1 = 0\), a contradiction against the supposition that \(m n\) such that \(m, n \lt m n\) was the smallest such.
So, \(Ch (R)\) is a prime number.
4: Note 2
\((m \cdot 1) (n \cdot 1)\) and \(m \cdot (n \cdot 1)\) mean some different things, but they have the same value after all because of the distribution law: \((m \cdot 1) (n \cdot 1) = (1 + ... + 1) (1 + ... + 1)\) while \(m \cdot (n \cdot 1) = (1 + ... + 1) + ... + (1 + ... + 1)\).
