  It is proved that an element $r$ in the center of a coherent ring $\Lambda$
annihilates $\mathrm{Ext}^{n}_{\Lambda}(M,N)$, for some positive integer $n$
and all finitely presented $\Lambda$-modules $M$ and $N$, if and only if the
bounded derived category of $\Lambda$ is an extension of the subcategory
consisting of complexes annihilated by $r$ and those obtained as $n$-fold
extensions of $\Lambda$. This has applications to finiteness of dimension of
derived categories.
