definition of characteristic of ring
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of characteristic of 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:
\( R\): \(\in \{\text{ the rings }\}\)
\(*Ch (R)\): \(\in \mathbb{N}\)
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Conditions:
\(Ch (R) \cdot 1 = 0 \land \forall n \in \mathbb{N} \setminus \{0\} \text{ such that } n \lt Ch (R) (n \cdot 1 \neq 0)\)
\(\lor\)
\(Ch (R) = 0\)
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2: Note
In other words, \(Ch (R)\) is the smallest positive natural number such that \(Ch (R)\) times \(1\) is \(0\) if there is such, or \(0\) if there is no such.
\(Ch (R) = 1\) means that \(1 \cdot 1 = 1 = 0\), which implies that \(R = \{0 = 1\}\), because for each \(r \in R\), \(r = r 1 = r 0 = 0\).
