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If      Λ  \Lambda       is an artin algebra there is a partition of      ind ⁡  Λ  \operatorname {ind} \Lambda      , the category of indecomposable finitely generated      Λ  \Lambda      -modules,      ind ⁡  Λ  = ∪  i ⩾  0 P _  _  i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i}      , called the preprojective partition. We show that      P _  _  i \underline {\underline {P}}_i       can be easily constructed for hereditary artin algebras, if      P _  _  i −  1 \underline {\underline {P}}_{i - 1}       is known:      A A       is in      P _  _  i \underline {\underline {P}}_i       if and only if      A A       is not in      P _  _  i −  1 \underline {\underline {P}}_{i - 1}       and there is an irreducible map      B →  A B \to A      , where      B B       is in      P _  _  i −  1 \underline {\underline {P}}_{i - 1}      .

A note on preprojective partitions over hereditary Artin algebras