Characteristic Times Element of Ring is Zero

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Theorem

Let $\left({R, +, \circ}\right)$ be a ring with unity.

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

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

Then:

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

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

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

Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle n \cdot a$	$=$	$\displaystyle n \cdot \left({1_R \circ a}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of unity
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left({n \cdot 1_R}\right) \circ a$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Powers of Ring Elements
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 0_R \circ a$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of characteristic
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle 0_R$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of ring zero

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle n \cdot a\)	\(=\)	\(\displaystyle n \cdot \left({1_R \circ a}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of unity
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({n \cdot 1_R}\right) \circ a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Powers of Ring Elements
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0_R \circ a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of characteristic
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0_R\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of ring zero