Diagonal F-splitting and Symbolic Powers of Ideals

Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$ essentially of finite type over an $F$-finite field. We show that $J^{s+t}\subseteq \tau (J^{s-\epsilon })\tau (J^{t-\epsilon })$ for all $s,t,\epsilon >0$ for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that $P^{(2hn)}\subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.