  Consider a reductive linear algebraic group $G$ acting linearly on a
polynomial ring $S$ over an infinite field; key examples are the general linear
group, the symplectic group, the orthogonal group, and the special linear
group, with the classical representations as in Weyl's book: for the general
linear group, consider a direct sum of copies of the standard representation
and copies of the dual; in the other cases take copies of the standard
representation. The invariant rings in the respective cases are determinantal
rings, rings defined by Pfaffians of alternating matrices, symmetric
determinantal rings, and the Pl\"ucker coordinate rings of Grassmannians; these
are the classical invariant rings of the title, with $S^G\subseteq S$ being the
natural embedding.
  Over a field of characteristic zero, a reductive group is linearly reductive,
and it follows that the invariant ring $S^G$ is a pure subring of $S$,
equivalently, $S^G$ is a direct summand of $S$ as an $S^G$-module. Over fields
of positive characteristic, reductive groups are typically no longer linearly
reductive. We determine, in the positive characteristic case, precisely when
the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$
is pure, then either the invariant ring $S^G$ is regular, or the group $G$ is
linearly reductive.
