TITLE: Sum of radical ideals
QUESTION [9 upvotes]: 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 $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$.

REPLY [4 votes]: With regards to the original question — I don't have any good ideas, probably valuation rings satisfy this (but they are not generally Noetherian except in the cases already outlined).
With regards to other classes of ideals that satisfy this (now in characteristic zero), the ideals of “unions of log canonical centers” satisfy this property in characteristic zero (this is mostly due to a result of Florin Ambro I think).  See for example Ambro - Basic properties of log canonical centers.