  We study primitive ideals in the enveloping algebra of finitary locally
finite infinite-dimensional complex Lie algebras. In particular we investigate
the annihilators of the simple objects in the category of tensor modules. This
category has been studied in \cite{PStyr} and \cite{PS}. Moreover we prove that
a simple tensor module is completely determined by its annihilator and we
introduce a partial order on pairs of Young diagrams that we use to describe
the inclusions between the annihilators of the simple tensor modules.
