The $H$-space that represents Brown--Peterson cohomology $\BP^k (-)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P(n)$, by constructing idempotent operations in $P(n)$-cohomology $P(n)^k ($--$)$ in the style of Boardman--Johnson--Wilson; this relies heavily on the Ravenel--Wilson determination of the relevant Hopf ring. The resulting $(i- 1)$-connected $H$-spaces $Y_i$ have free connective Morava $\K$-homology $k(n)_* (Y_i)$, and may be built from the spaces in the $\Omega$-spectrum for $k(n)$ using only $v_n$-torsion invariants. We also extend Quillen's theorem on complex cobordism to show that for any space $X$, the \linebeak$P(n)_*$-module $P(n)^* (X)$ is generated by elements of $P(n)^i (X)$ for $i \ge 0$. This result is essential for the work of Ravenel--Wilson--Yagita, which in many cases allows one to compute $\BP$-cohomology from Morava $\K$-theory.