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We show that the finite dimensional nilpotent complex Lie superalgebras gwhose injective hulls of simple U(g)-modules are locally Artinian are preciselythose whose even part g_0 is isomorphic to a nilpotent Lie algebra with anabelian ideal of codimension 1 or to a direct product of an abelian Lie algebraand a certain 5-dimensional or a certain 6-dimensional nilpotent Lie algebra.

Injective hulls of simple modules over finite dimensional nilpotent complex Lie superalgebras