Frobenius splitting and smoothness Let $R\rightarrow S$ be a morphism of rings in characteristic $p$ which is formally smooth. 
Is it true that $R$ is Frobenius splitting if and only if $S$ is Frobenius splitting?
One direction seems to be easy but I am not sure if both implications are true.
 A: Consider the following.  I'm going to assume that all rings are Noetherian (some things probably generalize, but I want to be careful). First a definition, 
Definition: An extension of rings $A \subseteq B$ is pure if $M \otimes_A A \to M \otimes_A B$ is injective for every $A$-module $M$.  A ring $A$ is called $F$-pure if the Frobenius map $F : A \to A$ is pure.  For an $F$-finite ring, being $F$-pure is the same as being split.  
Hochtser-Roberts, The purity of the Frobenius and local cohomology, Proposition 5.4(b) implies that if $A \to B$ is faithfully flat, then it's pure.  I'm going to then assume that in your case, the ring maps are faithfully flat.  
Theorem: (Hochster-Roberts, Prop 5.4(a))  If $A \subseteq B \subseteq C$ are rings and $B$ is pure in $C$, then $A$ is pure in $B$ if and only if $A$ is pure in $C$.
Now suppose that $R \subseteq S$ is faithfully flat (I'm going to assume you can do faithful, as otherwise there isn't any hope).
Ok, so then set $A = R$, $B=S$ and $C = S$ with $B \to C$ being the Frobenius on $S$.  Likewise set $B' = R$ with $R = A \to B'$ being the Frobenius on $R$.  In otherwords we have
$$
A \subseteq B \subseteq C
$$
and
$$
A \subseteq B' \subseteq C
$$
Since $S$ is $F$-pure, $B \to C$ is pure, and so $A \to C$ is pure since $A \to B$ is faithfully flat.  But now certainly $B' \to C$ is also pure (since it is also faithfully flat) and so $A \to B'$ is also pure so that $R = A$ is $F$-pure.
Conclusion: If $S$ is $F$-pure and $R \subseteq S$ is faithfully flat, then $R$ is $F$-pure.

For the other direction, obviously faithful flatness will not be good enough but the following is true.
Theorem: (Hochster-Huneke?) If $(R, \mathfrak{m}) \subseteq (S, \mathfrak{n})$ is a flat local extension of local rings (say excellent to be safe) such that the closed fiber is regular and $R$ is $F$-pure, then $S$ is $F$-pure.
Hochster and Huneke proved a harder result (replacing $F$-purity by $F$-regularity in F-Regularity, Test Elements, and Smooth Base Change Theorem 7.3 and adding some assumptions about generic fibers which allow one to mess about with test elements, see the theorem on page 171 of these notes by Hochster)  Their methods will prove the Theorem.  The case you probably want follows directly from the proof of a result of myself and Wenliang Zhang, Lemma 4.5 in Bertini Theorems for $F$-singularities, but the above theorem was certainly known long before that by all the experts.  
