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?

usefulnotion of primary decomposition for coherent sheaves $\mathcal{F}$ on locally noetherian schemes $X$ (recovering my suggestion for the structure sheaf). Beware of two typos: in 3.2.5 the quotients $\mathcal{F}_ {\alpha}$ of $\mathcal{F}$ should be required to be quasi-coherent (equivalently, coherent), and more importantly in 3.2.6 at the end $\kappa(x)$ should be $\mathcal{O}_ {X,x}$. Hopefully my preceding comment clarifies things, but seriously, just read EGA. $\endgroup$ – Boyarsky Jul 7 '10 at 4:06