  Let S(n) be the category of invariant subspaces of nilpotent operators with
nilpotency index at most n. Such submodule categories have been studied already
in 1936 by Birkhoff, they have attracted a lot of attention in recent years,
for example in connection with some weighted projective lines (Kussin, Lenzing,
Meltzer). On the other hand, we consider the preprojective algebra of type A_n;
the preprojective algebras were introduced by Gelfand and Ponomarev, they are
now of great interest, for example they form an important tool to study quantum
groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schroeer). Direct
connections between the submodule category S(n) and the module category of the
preprojective algebra of type A_{n-1} have been established quite a long time
ago by Auslander and Reiten, and recently also by Li and Zhang, but apparently
this remained unnoticed. The aim of this note is to provide details on this
relationship. As a byproduct we see that here we deal with ideals I in
triangulated categories T such that I is generated by an idempotent and T/I is
abelian.
