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Revision difference : primal element |
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Version 4 |
| An element $r$ in a commutative ring $R$ is called \emph{primal} if whenever $r\mid ab$, with $a,b\in R$, then there |
An element $r$ in a commutative ring $R$ is called \emph{primal} if whenever $r\mid ab$, with $a,b\in R$, then there |
| exist elements $s,t\in R$ such that |
exist elements $s,t\in R$ such that |
| \begin{enumerate} |
\begin{enumerate} |
| \item $r=st$, |
\item $r=st$, |
| \item $s\mid a$ and $t\mid b$. |
\item $s\mid a$ and $t\mid b$. |
| \end{enumerate} |
\end{enumerate} |
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\textbf{Lemma}. In a commutative ring, an element that is both irreducible and primal is a prime element.
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\textbf{Remark}. In a commutative ring, an element that is both irreducible and primal is a prime element.
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| \begin{proof} |
\begin{proof} |
| Suppose $a$ is irreducible and primal, and $a\mid bc$. Since $a$ is primal, there is $x,y\in R$ such that $a=xy$, with $x\mid b$ and $y\mid c$. Since $a$ is irreducible, either $x$ or $y$ is a unit. If $x$ is a unit, with $z$ as its inverse, then $za=zxy=y$, so that $a\mid y$. But $y\mid c$, we have that $a\mid c$. |
Suppose $a$ is irreducible and primal, and $a\mid bc$. Since $a$ is primal, there is $x,y\in R$ such that $a=xy$, with $x\mid b$ and $y\mid c$. Since $a$ is irreducible, either $x$ or $y$ is a unit. If $x$ is a unit, with $z$ as its inverse, then $za=zxy=y$, so that $a\mid y$. But $y\mid c$, we have that $a\mid c$. |
| \end{proof} |
\end{proof} |
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