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Version 2 |
| \PMlinkescapeword{ideals} \PMlinkescapeword{ideal} |
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| \textbf{Lemma.}\, Let $R$ be a commutative ring and $S$ a multiplicative semigroup consisting of a subset of $R$.\, If there exist \PMlinkid{ideals}{371} of $R$ which are disjoint with $S$, then the set $\mathfrak{S}$ of all such ideals has a maximal element with respect to the set inclusion. |
\textbf{Lemma.}\, Let $R$ be a commutative ring and $S$ a multiplicative semigroup consisting of a subset of $R$.\, If there exist \PMlinkid{ideals}{371} of $R$ which are disjoint with $S$, then the set $\mathfrak{S}$ of all such ideals has a maximal element with respect to the set inclusion. |
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| {\em Proof.}\, Let $C$ be an arbitrary chain in $\mathfrak{S}$.\, Then the union |
{\em Proof.}\, Let $C$ be an arbitrary chain in $\mathfrak{S}$.\, Then the union |
| $$\mathfrak{b} \;:=\; \bigcup_{\mathfrak{a} \in C}\mathfrak{a},$$ |
$$\mathfrak{b} \;:=\; \bigcup_{\mathfrak{a} \in C}\mathfrak{a},$$ |
| which belongs to $\mathfrak{S}$, may be taken for the upper bound of $C$, since it clearly is an ideal of $R$ and disjoint with $S$.\, Because $\mathfrak{S}$ thus is inductively ordered with respect to ``$\subseteq$'', our assertion follows from Zorn's lemma.\\ |
which belongs to $\mathfrak{S}$, may be taken for the upper bound of $C$, since it clearly is an ideal of $R$ and disjoint with $S$.\, Because $\mathfrak{S}$ thus is inductively ordered with respect to ``$\subseteq$'', our assertion follows from Zorn's lemma.\\ |
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| \textbf{Definition.}\, The maximal elements in the Lemma are {\em prime ideals} of the commutative ring.\\ |
\textbf{Definition.}\, The maximal elements in the Lemma are {\em prime ideals} of the commutative ring.\\ |
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| The ring $R$ itself is always a prime ideal ($S = \varnothing$).\, If $R$ has no zero divisors, the zero ideal $(0)$ is a prime ideal ($S = R\!\smallsetminus\!\{0\}$). |
The ring $R$ itself is always a prime ideal ($S = \varnothing$).\, If $R$ has no zero divisors, the zero ideal $(0)$ is a prime ideal ($S = R\!\smallsetminus\!\{0\}$). |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Artin} {\sc Emil Artin}: {\em Theory of Algebraic Numbers}.\, Lecture notes.\, Mathematisches Institut, G\"ottingen (1959). |
\bibitem{Artin} {\sc Emil Artin}: {\em Theory of Algebraic Numbers}.\, Lecture notes.\, Mathematisches Institut, G\"ottingen (1959). |
| \end{thebibliography} |
\end{thebibliography} |
