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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 2^\circ$