The
$H$
-space that represents Brown-Peterson cohomology
$\text{B}{{\text{P}}^{k}}\left( - \right)$
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
$P\left( n \right)$
, by constructing idempotent operations in
$P\left( n \right)$
-cohomology
$P{{(n)}^{k}}\left( - \right)$
in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting
$\left( i\,-\,1 \right)$
-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
$k\left( n \right)$
using only
${{v}_{n}}$
-torsion invariants.
We also extend Quillen's theorem on complex cobordism to show that for any space
$X$
, the
$P{{\left( n \right)}_{*}}$
-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.