I am looking for a reference/proof that shows that the Morava $K$-theory spectra, $K(n)$ are not $E\_{\infty}$ ring spectra. I suspect that this should be a calculation but I can't quite get it right.


Thank you as always.



Morava $K(n)$'s are not $E_{\infty}$
The assertion is Lemma 5.6.4 in Rognes's "Galois extensions of structured ring spectra" available on the [arXiv](http://arxiv.org/abs/math/0502183). In fact, the $K(n)$ spectra do not even admit $E\_2$-algebra structures. The reason is that the free $E\_2$-algebra with $p = 0$ is known to be (by a theorem of Hopkins-Mahowald) the Eilenberg-MacLane spectrum $H \mathbb{F}\_p$. If $K(n)$ admitted an $E\_2$-algebra structure, it would receive a map of $E\_2$-rings $H \mathbb{F}\_p \to K(n)$, which is not possible (for instance, as $H \mathbb{F}\_p \wedge K(n) = 0$). 


More generally, no $p$-power torsion $E\_n$-local spectrum can admit an $E\_\infty$-structure. This can be seen by playing with power operations and is worked out as the main result of [joint work](http://arxiv.org/abs/1403.2023) with J. Noel and N. Naumann. The closing remarks to that paper also outline a proof of the Hopkins-Mahowald theorem cited above (which was not formally documented in the literature for odd primes). 


Analogous results for $E\_2$-algebras (or $E\_m$-algebras for $m \geq 2$) seem to be unknown. For example, consider the free $E\_2$-algebra with $p^2 =0 $. Is it true that this is $K(n)$-acyclic for each $n$? 


At the prime $2$, it is even true that any *homotopy commutative* ring spectrum with $2 =0$ is equivalent to a wedge of Eilenberg-MacLane spectra (and in particular is invisible to $L\_n$-localization). This is a [result](http://link.springer.com/article/10.1007%2FBF02621900#page-2) of Wurgler. 


