# Characteristic times Ring Element is Ring Zero

## 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:

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

$\blacksquare$