The wedge sum $\bigvee\_{k \in 2 \mathbb{Z}} S^{k}$ is an $A\_\infty$-ring spectrum: the connective cover is the free $A\_\infty$-ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the element in $\pi\_2$. It is also homotopy commutative. Can it be made into an $E\_\infty$-ring spectrum? 
More generally, given an $E\_\infty$-ring spectrum $R$, when can $\bigvee\_{k \in 2 \mathbb{Z}} \Sigma^k E$ be made into an $E\_\infty$-ring? 



Is a wedge of spheres an $E_\infty$ ring spectrum? 
Ah, tracked it down. Here is an argument. I should mention that Peter once pointed me towards an original source due to McClure, or page 238, Prop. 6.1, of SLN 1176 (the $H\_\infty$ book).


Suppose you had such a ring object $R$. We examine its mod-2 homology. This has several features:


* It is the ring $\mathbb Z/2[t^{\pm 1}]$.
* It has trivial action of the Steenrod operations $P\_r$ (this is dual to the cohomology action), because it's a wedge of spheres.
* From the $H\_\infty$ book, it has Dyer-Lashof operations $Q^s$. These satisfy $Q^{|x|} x = x^2$, and the Nishida relations
$$
P\_r Q^s = \sum \binom{s - r}{r - 2i} Q^{s-r+i}P\_i.
$$


In particular, these together would say 
$$
0 = P\_2 Q^4 t = \binom{2}{2} Q^{2}P\_0 t + \binom{2}{0} Q^{4}P\_2 t = Q^2 t = t^2.
$$


If $E$ is a commutative $MU$-algebra, then you can use the map from $MU$ to its periodic version $MUP$ to produce $R \wedge\_{MU}MUP$, which is 2-periodified. Unfortunately, we know very few genuine $MU$-algebras. Barry Walker proved that complex K-theory is one of them, based on Matthew Ando's work on studying $H\_\infty$ structures on Lubin-Tate cohomology theories. My understanding is that the problem is still open for almost all of the Lubin-Tate cohomology theories.


