  The $Z\to \gamma \gamma$ and $Z\to gg$ decays are studied in the context of
the renormalizable version of the Standard Model Extension. The $CPT$-odd
$\bar{\psi}\gamma_5 \gamma^\mu b_\mu\psi$ bilinear interaction, which involves
the constant background field $b_\alpha$ and which has been a subject of
interest in literature, is considered. It is shown that the $Z\to \gamma
\gamma$ and $Z\to gg$ decays, which are strictly zero in the standard model,
can be generated radiatively at the one-loop level. It is found that these
decays are gauge invariant and free of ultraviolet divergences, and that the
corresponding decay widths only depend on the spatial component of the
background field $b$.
