Do there exist "topologically significant" (and not "algebraic") triangulated categories killed by the multiplication by $p$? I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably correspond to the study of $p$-torsion of (co)homology. As far as I remember, $S^0/p$ is not a ring spectrum by a result of Schwede (so, one cannot consider modules over it); yet does there exist any method for "overcoming" this difficulty (and still obtaining a "non-algebraic" triangulated category) somehow? Here non-algebraic means that the category should not be "closely related" to the derived category of any abelian category. I suspect that there should exist certain categories of this sort since certain cohomology theories (including $K$-theory) cannot be defined in terms of complexes.
 A: Depending on exactly what you mean by "killed by $p$", the answer may be no.
Let $\mathcal{C}$ be a stable $\infty$-category and let $\iota_{\mathcal{C}}$ be the identity functor from $\mathcal{C}$ to itself. If the ``multiplication by $p$'' map
$p: \iota_{\mathcal{C}} \rightarrow \iota_{\mathcal{C}}$ is nullhomotopic, then
$\mathcal{C}$ is $\mathbf{F}_p$-linear (and can therefore be obtained from a pretriangulated differential graded category over $\mathbf{F}_p$). 
A: Yes. For $p$ a prime and $n > 0$, the Morava $K$-theories $K(n)$ and their connective versions $k(n)$ are associative (but not commutative) ring spectra with coefficient rings $\Bbb F_p[v_n^{\pm 1}]$ and $\Bbb F_p[v_n]$ respectively, and the homotopy categories of their module categories are triangulated categories in which $p=0$. These homotopy categories are equivalent to the derived categories of modules over their coefficient rings (this was studied by Franke and in this paper of Patchkoria, and is roughly a consequence of the coefficient rings having small homological dimension).
There has been quite a lot of work on this recently, though I'm having trouble finding references to more recent developments. I believe that it is still an open question whether these equivalences of categories lift to equivalences of triangulated categories, but that this problem is most difficult for $p$ and $n$ small.
(There are further variants of this for which we let the finite field, or a formal group law over it, vary.)
