let $A$ be a ring and endow the closed subsets of $Spec A$ with the grothendieck topology of finite covers. it would be nice if $V \mapsto A/I(V)$ defines a sheaf. this is not true in general and is related (equivalent?) to the following pure algebraic question:
in which rings $A$ are the radical ideals closed under sum?
if $A$ is a noetherian integral domain of dimension $0$ or $1$, it's true. you cannot omit integral here (consider $k[x,y]/(x^2 y + y^2)$ localized at $(x,y)$), nor the dimension (in $k[x,y]$, consider $(x^2 + y)+(y) = (x^2,y)$). are there other interesting examples/counterexamples or approaches for general classification?