I am reading Thomason's "The Classification of Triangulated Subcategories". There we learn that for a given $\otimes$-triangulated category $\mathcal T$ and a subset of objects $E\subseteq \mathcal T$ the smallest thick $\otimes$-triangulated subcategory $\langle E \rangle$ is obtained by taking the intersection of all such subcategories which contain $E$.
Therefore the collection of all thick $\otimes$-triangulated subcategories form a complete lattice, we will call it Ideal$(\mathcal T)$. Since each ideal is closed under summands, I would think this structure would differ from its ring theoretic analogue. For example, I am wondering
If $F\colon \mathcal T\to \mathcal T'$ is a $\otimes$-triangulated functor, does $F^{-1}\colon$Ideal$(\mathcal T')\to$Ideal$(\mathcal T)$ preserve joins?
This is clearly not the case for ring morphisms, since $f\colon k[T]\to k[X,Y]$, $T\mapsto X+Y$ has $f^{-1}(X)+f^{-1}(Y)=0\neq (T)=f^{-1}((X)+(Y))$ (Note that we are working on ideals here).
What I am really looking for is to have $F^{-1}$ preserve joins of prime ideals (those $\mathfrak p$ which have $a\otimes b\in \mathfrak p$ iff $a\in \mathfrak p$ or $b\in \mathfrak p$).