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