Allen Knutson has a nice recent preprint [Frobenius splitting, point-counting, and degeneration][1] which, among other things, discusses a class of rings (of prime characteristic) for which a certain <strike>super</strike>subclass 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 $I+J$.


  [1]: https://arxiv.org/abs/0911.4941