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On a related note, however, there are some important classes of rings $R$ where \emph{countable countable prime avoidance} holds. That is, when an ideal $J$ is a subset of a \emph{countable} countable union of prime ideals in such a ring, it has to be contained in a single element of the collection. In particular, $R$ satisfies countable prime avoidance if either:

  1. $R$ contains an uncountable field, or

  2. $R$ is a complete Noetherian local ring.

Case 1 was mentioned in an article by Hochster and Huneke in the Michigan Math. Journal in 2000. Case 2 was proved by Lindsay Burch in 1972. Both arguments are fairly straightforward.

I learned these things from the Hochster-Huneke article some years back.
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On a related note, however, there are some important classes of rings $R$ where \emph{countable prime avoidance} holds. That is, when an ideal $J$ is a subset of a \emph{countable} union of prime ideals in such a ring, it has to be contained in a single element of the collection. In particular, $R$ satisfies countable prime avoidance if either:\begin{enumerate} \item

  1. $R$ contains an uncountable field, or\item

  2. $R$ is a complete Noetherian local ring.\end{enumerate}

Case 1 was mentioned in an article by Hochster and Huneke in the Michigan Math. Journal in 2000. Case 2 was proved by Lindsay Burch in 1972. Both arguments are fairly straightforward.

I learned these things from the Hochster-Huneke article some years back.
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On a related note, however, there are some important classes of rings $R$ where \emph{countable prime avoidance} holds. That is, when an ideal $J$ is a subset of a \emph{countable} union of prime ideals in such a ring, it has to be contained in a single element of the collection. In particular, $R$ satisfies countable prime avoidance if either: \begin{enumerate} \item $R$ contains an uncountable field, or \item $R$ is a complete Noetherian local ring. \end{enumerate} Case 1 was mentioned in an article by Hochster and Huneke in the Michigan Math. Journal in 2000. Case 2 was proved by Lindsay Burch in 1972. Both arguments are fairly straightforward.

I learned these things from the Hochster-Huneke article some years back.