Characteristic of ring equals additive order of unity
Dependencies:
Let $R$ be a ring with unity.
The characteristic of $R$ ($\operatorname{char}(R)$) is the smallest positive integer $n$ such that $\forall r \in R, nr = 0$. If no such $n$ exists, $\operatorname{char}(R) = 0$.
$\operatorname{char}(R) = 0 \iff \operatorname{order_+}(1) = \infty$. Otherwise $\operatorname{char}(R) = \operatorname{order_+}(1)$.
Proof
Let $n = \operatorname{char}(R)$ and $m = \operatorname{order_+}(1)$.
Lemma 1
Let $k \in \mathbb{Z}$.
\begin{align} kr &= (r + r + \ldots + r) \\ &= (r \cdot 1 + r \cdot 1 + \ldots + r \cdot 1) \\ &= r \cdot (1 + 1 + \ldots + 1) \tag{distributivity} \\ &= r \cdot (k1) \end{align}
Therefore, $kr = r(k1)$.
Part 1
Assume $m \neq \infty$. Then $mr = r(m1) = r0 = 0$. Therefore, characteristic exists, which means $n \neq 0$.
Assume $n \neq 0$. Then $n1 = 0 \Rightarrow m \mid n$. Therefore $m \neq \infty$.
Therefore, $n = 0 \iff m = \infty$.
Part 2
Assume $n \neq 0$ (which means $m \neq \infty$).
Since $n1 = 0$, $m \mid n \Rightarrow m \le n$.
$mr = r(m1) = r0 = 0$. Therefore, $n \le m$. Therefore, $m = n$.
Dependency for: None
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- Depth: 2
- Number of transitive dependencies: 2