Let $A$ be a commutative ring and endow the closed subsets of $mathrm{Spec}(A)$ 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 SE/322872.
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