Multiple Function on Ring is Zero iff Characteristic is Divisor

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

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


Let $a \in R$ such that $a$ is not a zero divisor of $R$.

Then:

$n \cdot a = 0_R$

if and only if:

$p \divides n$

where $\cdot$ denotes the multiple operation.


Proof

Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:

$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$

Then from Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic:

$\map \ker {g_a} = \ideal p$

where:

$\map \ker {g_a}$ is the kernel of $g_a$
$\ideal p$ is the principal ideal of $\Z$ generated by $p$.

We have by definition of kernel:

$n \in \map \ker {g_a} \iff n \cdot a = 0_R$

and by definition of principal ideal:

$n \in \ideal p \iff p \divides n$

The result follows.

$\blacksquare$


Sources