  Following the article of C. M. Ringel we introduce preprojective algebras of
a Dynkin quiver $Q$ starting from three definitions which, despite concerning
completely different algebraic structures, turn out to be equivalent. Our main
result is a new version of Ringel's proofs that applies a theorem by Happel and
exploits the techniques of homological algebra. Moreover we show that the
definition of the preprojective algebra given with the usual notion of
commutator is equivalent to the definition with the "generalised" commutator.
