  We give an elementary proof of the Gel'fand-Kapranov-Zelevinsky theorem that
non-resonant A-hypergeometric systems are irreducible. We also provide a proof
of a converse statement
  In this second version we have removed the condition of saturatedness in
Theorems 1.2 and 1.3. In Theorem 1.3 it is replaced by the condition of
Cohen-Macaulayness of the toric ideal.
