Is it known what conditions we require of a stable homotopy category to have $\langle S\rangle = \coprod\limits_{\mathbb{N}}\langle K(n)\rangle$, where $\langle K(n)\rangle$ is some minimal Bousfield class? That is, more specifically, in which cases can we say that Proposition 3.7.5 of Hovey, Palmieri and Strickland's "Axiomatic Stable Homotopy Theory" applies?
