Extension of irreducible ideal of a Noetherian ring is irreducible of rings of fractions

Question

Let $S$ be a multiplicatively closed subset of the commutative Noetherian ring $R$ and $f:R \to {S^{ - 1}}R$ denote the natural ring homomorphism.

Let $I$ be an irreducible ideal of $R$ for which $S \cap I = \emptyset $. Show that $I^e$ is an irreducible ideal of ${S^{ - 1}}R$.

(Where $I^e$ is extension ideal of $I$ and $I$ is irreducible if and only if $I ≠R$ and, whenever $I=I_1∩I_2$, with $I_1$ and $I_2$ ideals of $R$, then $I=I_1$ or $I=I_2$)

I have just proved that if $I$ is an irreducible ideal of ${S^{ - 1}}R$ then $I^c$ is an irreducible ideal of $R$. I don't know how to use the Noetherian condition to prove the opposite side.

Answer 1

Suppose that $I^e = J_1 \cap J_2$ for distinct proper $S^{-1}R$-ideals $J_i$, then $I^{ec} = (J_1 \cap J_2)^c = J_1^c \cap J_2^c$. Note that $J_1^c$ and $J_2^c$ are distinct proper ideals of $I$ because for localization maps, $J^{ce} = J$ (more generally this is true of flat epimorphisms).

Thus if we demonstrate that $I^{ec} = I$ under the conditions on $R$ and $I$, then we will have showed the desired implication "$I^e$ reducible $\implies$ $I$ reducible."

Noetherian rings have primary decompositions, so $I$ being irreducible implies its being primary. All that remains is to understand how primary ideals behave under localization. The answer is basically that they behave analogously to prime ideals, as well as one could hope for.

Lemma Let $I$ be a primary ideal of a commutative ring $R$ and $R \rightarrow S^{-1}R$ localization at a multiplicative set $S$. Extension and contraction of ideals induces a one-to-one correspondence between primary ideals of $S^{-1}R$ and primary ideals of $R$ disjoint from $S$. In particular if $I$ is a primary ideal of $R$ disjoint from $S$ then $I^e$ is primary ideal of $S^{-1}R$ and $I^{ec} = I$.

I leave the details of this to you.

Note that we did not need the full power of being Noetherian here. All we used was the implication "irreducible $\implies$ primary" which holds for Noetherian rings, and more generally Laskerian rings (in which primary decompositions exist).