In his great answer to this MO question James Borger gives a geometric characterization of non-unital $k$-algebras (i.e. not necessarily unital) : They correspond to affine schemes $X$ over $k$ together with a morphism $\text{Spec}(k) \to X$ over $k$. More generally, what about defining a *non-unital* scheme over a base scheme $S$ as a usual scheme $X$ over $S$ together with a morphism $S \to X$ over $S$? The idea is that this section "cuts out the unital part" of $X$.

**Question:** In the same spirit as algebraic geometry answers questions about commutative algebra and vice versa, is there some non-unital algebraic geometry (perhaps with the above definition of non-unital schemes) which interacts with non-unital commutative algebra? What are explicit applications?

See this MO question for some important examples of non-unital rings. So my question is basically if we can study them via some kind of spectral theory. Note that the naive approach, just imitating the unital case by considering the set of prime ideals, does not work at all since there are not enough prime ideals to establish some basic facts about the structure sheaf, even if we assume something like $A*A = A$.

For non-unital rings in which every element is idempotent, i.e. boolean rings, the spectral theory should yield the non-unital Stone duality out of the unital Stone duality. Likewise with some modifications (restriction to $\mathbb{C}$-points) we should get the non-unital Gelfand duality out of the unital Gelfand duality.