Prime thick subcategories and spectra of derived and singularity categories of noetherian schemes

For an essentially small triangulated category $\mathcal{T}$, we introduce the notion of prime thick subcategories and define spectrum which shares basic properties with a tensor introduced by Balmer. We mainly focus on categories that appear in algebraic geometry such as derived singularity noetherian scheme $X$. prove certain classes are these categories. Furthermore, use this result to show subspaces $X$ embedded into their spectra topological spaces.