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

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

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

where $\cdot$ denotes the multiple operation.


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

Then:

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

where:

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


Proof

From Principal Ideal of Characteristic of Ring is Subset of Kernel of Multiple Function we have:

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

for all $a \in R$.

It remains to be shown that for all $a \in R$ such that $a$ is not a zero divisor of $R$:

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


So:

\(\ds n\) \(\in\) \(\ds \map \ker {g_a}\)
\(\ds \leadsto \ \ \) \(\ds n \cdot a\) \(=\) \(\ds 0_R\)
\(\ds \leadsto \ \ \) \(\ds n \cdot \paren {a \circ 1_R}\) \(=\) \(\ds 0_R\) Definition of Unity of Ring
\(\ds \leadsto \ \ \) \(\ds \paren {n \cdot a} \circ 1_R\) \(=\) \(\ds 0_R\) Multiple of Ring Product
\(\ds \leadsto \ \ \) \(\ds a \circ \paren {n \cdot 1_R}\) \(=\) \(\ds 0_R\) Multiple of Ring Product
\(\ds \leadsto \ \ \) \(\ds n \cdot 1_R\) \(=\) \(\ds 0_R\) as $a$ is not a zero divisor of $R$
\(\ds \leadsto \ \ \) \(\ds n\) \(\in\) \(\ds \ideal p\) Definition 2 of Characteristic of Ring

Hence the result by definition of subset.

$\blacksquare$


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