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We obtain an improvement of some coloring theorems from \cite{nsbpr},\cite{819}, and \cite{APAL} for the case where the singular cardinal inquestion has countable cofinality. As a corollary, we obtain an "idealized"version of the combinatorial principle $\pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$ thatmaximizes the indecomposability of the associated ideal.

On idealized versions of $\pr_1(\mu^+,\mu^+,\mu^+,\cf(\mu))$