Affine schemes correspond to unitial commutative rings, of course.
Further, let us draw upon the Gelfand correspondence, where commutative $C^*$-algebras are dual to compact hausdorff spaces. If we lax the condition that these $C^*$-algebras be unitial, then then the correspondence is with locally compact hausdorff spaces instead. The functionals on a locally compact hausdorff space are required to have compact support.
My question:
Is there a way of viewing general schemes as corresponding to unitless rings, in a way extending the correspondence between rings and affine schemes?
The proposed correspondence would be this: write a scheme $X$ as a filtered colimit of affine schemes $\text{colim}_I X_i$. Then consider $\text{lim}_I \mathcal{O}_{X_i}(X_i)$ in the category of $\textit{unitless}$ rings.
I have slight suspicions that this approach works, given that projective space has simply a field as its global sections.
Perhaps there is some necessary condition on the maps, such as locally unit preserving. And for that matter, what is the new definition of prime ideal? It would seem that we want all primes to be proper still, but usually that is related to $1 \notin \mathfrak{p}$, so this may change.
But to be clear what would change is that now we consider sheaves in nonunitial commutative rings, and we make a sheaf out of such rings on their spectrum of primes (let's try the same definition, and tweak if we have to) all spaces are required to be affine. There should be a canonical projection of sheaves of nonunitial rings from the new structure sheaf to the usual structure sheaf. For projective space, this would be the projection $\text{limit} \mathcal{O}_{X_i}(X_i) \rightarrow k$, where this limit is taken of nonunitial rings.