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Homedefinition of prime ideal by Artin

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# definition of prime ideal by Artin

Lemma. Let $R$ be a commutative ring and $S$ a multiplicative semigroup consisting of a subset of $R$. If there exist ideals 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.

Proof. Let $C$ be an arbitrary chain in $\mathfrak{S}$. Then the union

$\mathfrak{b}\;:=\;\bigcup_{{\mathfrak{a}\in C}}\mathfrak{a},$ |

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.

Definition. The maximal elements in the Lemma are prime ideals of the commutative ring.

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 ($S=R\!\smallsetminus\!\{0\}$).

If the ring $R$ has a non-zero unity element 1, the prime ideals corresponding the semigroup $S=\{1\}$ are the maximal ideals of $R$.

# References

- 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).

## Mathematics Subject Classification

13C99*no label found*06A06

*no label found*

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