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