Multiple Function on Ring is Zero iff Characteristic is Divisor
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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:
- $n \cdot a = 0_R$
- $p \divides n$
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 2^\circ$