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Allen Knutson has a nice recent preprint Frobenius splitting, point-counting, and degeneration which, among other things, discusses a class of rings (of prime characteristic) for which a certain supersubclass of the radical ideals is closed under sum. They're called "Frobenius split rings." I guess they're originally defined by Brion and Kumar. They are defined to be those rings $R$ of characteristic $p$ admitting an additive map $\phi\colon R \rightarrow R$ such that $\phi(a^pb) = a\phi(b)$ and $\phi(1)=1$. An ideal $I$ is "compatibly split" if $\phi(I) \subseteq I$. Compatibly split ideals are radical, prime components of a compatibly split ideal are compatibly split, and if $I$ and $J$ are compatibly split, then so is $I+J$.