Let $A$ be a commutative ring and endow the closed subsets of $\operatorname{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:
In which commutative rings $A$ are the radical ideals closed under sum?
The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the $1$-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.
Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$.