Characteristic times Ring Element is Ring Zero

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity.

Let the zero of $R$ be $0_R$ and the unity of $R$ be $1_R$.

Let the characteristic of $R$ be $n$.


Then:

$\forall a \in R: n \cdot a = 0_R$


Proof

If $a = 0_R$ then $n \cdot a = 0_R$ is immediate.

So let $a \in R: a \ne 0_R$.

Then:

\(\displaystyle n \cdot a\) \(=\) \(\displaystyle n \cdot \paren {1_R \circ a}\) $\quad$ Definition of Unity of Ring $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {n \cdot 1_R} \circ a\) $\quad$ Powers of Ring Elements $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0_R \circ a\) $\quad$ Definition of Characteristic of Ring $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0_R\) $\quad$ Definition of Ring Zero $\quad$

$\blacksquare$


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