The Thom spectrum MO is a module over the ring spectrum π_{≤0}S=H**Z**, where S is the sphere spectrum.
In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ_{*}(MO).
On the other hand, MU and MSO are not modules over π_{≤0}S, because they have nontrivial k-invariants.

I wonder if the above result about MO can be generalized to other Thom spectra like MU and MSO
by considering higher truncations π_{≤k}S of the sphere spectrum.
Such a result would be interesting because it is related to the question of representing Thom spectra
as (weak) simplicial objects in the k-category of symmetric monoidal (k+1)-groups (k-groupoids with invertible objects), in the sense that
interpreting symmetric monoidal (k+1)-groups as stable homotopy k-types and taking the homotopy colimit should give back the Thom spectrum under consideration.
Such models are interesting because they are more strict then (say) Segal's Γ-spaces.
A negative answer would necessarily preclude the existence of such models because a stable homotopy k-type
is a module over π_{≤k}S, and the homotopy colimit of a simplicial diagram of modules over π_{≤k}S in the category of spectra is a again a module over π_{≤k}S.

**Is MU or MSO a module over the ring spectrum π _{≤k}S for some k>0? Same question for KU and KO.**

(Incidentally, the above result for MO implies that MO can be represented as a simplicial abelian group. I wonder if there is a geometric model for MO as a simplicial abelian group along the lines of Galatius-Madsen-Tillmann-Weiss theorem, i.e., n-simplices should be related to unoriented n-manifolds. Such a model would necessarily make explicit use of properties of unoriented manifolds as opposed to oriented or stably complex manifolds.)