I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call primary if for every non-empty open set $U$, the ring $\Gamma(U, \mathcal{O})$ is primary.
I will say that a scheme $X$ has a primary decomposition if there exist finitely many primary closed subschemes $Y_i$ such that the morphism $\phi \colon Y = \coprod_i Y_i \to X$ of schemes is surjective and, moreover, for every open $U \subset X$, the induced map $\Gamma(U, \mathcal{O}_X) \to \Gamma(\phi^{-1}(U), \mathcal{O}_Y)$ is injective. (Thus, a section is determined by its restrictions to the $Y_i$.) If I have not made any errors in setting up these definitions, then it is a standard theorem that every Noetherian affine scheme has a primary decomposition. However, these primary decompositions, even if we require them to be minimal, are not unique or canonical. Thus, if they can be glued together for a global construction, this is not immediately obvious.
Do primary decompositions exist for Noetherian schemes in general? If not, are there "reasonable" hypotheses (other than affineness) under which they do exist?
