Let $k$ be an algebraically closed field of characteristic p. Let
$Z\subset k[x_1,\cdots,x_n]$ be a $k$-subalgebra of a 
polynomial ring, such that for any $f\in Z,$ any divisor of $f$ (in
$k[x_1,\cdots,x_n]$) is again an element of $Z.$ Then is
$Z[x_1^p,\cdots,x_n^p]$ a normal ring? For the easiest case of
$Z=k[f]-$where $f$ is an irreducible polynomial, the answer is yes,
since $Z[x_1^p,\cdots,x_n^p]$ in this case is a complete intersection
and regular in codimension 2.