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Here is a slightly more fleshed out version of my comment. Let $K(1)$ be the first Morava $K$-theory. When $p$ is odd one can calculate the homotopy groups of the $K(1)$-localised sphere spectrum to b …

The real projective spaces $\mathbb{R}P^3 \cong SO(3)$ and $\mathbb{R}P^7$ also give fun examples. Naylor proved that there exist 768 $H$-space structures on $SO(3)$, while Rees shows that there exist …

In addition to the notes of Haynes Miller see http://jdc.math.uwo.ca/papers/ideals.pdf

Here is a slightly more fleshed out version of the comment above. First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of com …

Regarding $E(n)_*E(n)$, see "On the Structure of the Hopf Algebroid $E(n)_*E(n)$" by Keith Johnson. Johnson shows that $$E(n)_*E(n) \otimes \mathbb{Q} \simeq \mathbb{Q}[v_1,\cdots,v_{n-1},v_n^{\pm 1}, …

I've found a paper of Spanier's (Higher Order Operations) where he uses the theory of "carriers" to study $n$-th order operations. The set-up is rather general; for example a particular case defines t …

Just to close this off - thanks to Mike-Doherty it appears that the answer is yes (and in fact for spaces this goes back to the paper "The decomposition of stable homotopy" by Joel Cohen.) Using the …

To help with (3), let me point out that in 'nice' situations (e.g., in the derived category of a noetherian commutative ring), the Balmer spectrum classifies all localizing tensor ideals of the catego …