  Using a data sample of $(1310.6 \pm 7.0) \times 10^{6}$ $J/\psi$ events taken
with the BESIII detector at the center-of-mass energy of 3.097 GeV, we search
for the first time for the lepton number violating decay $\Sigma^{-} \to p
e^{-} e^{-}$ and the rare inclusive decay $\Sigma^{-} \to \Sigma^{+} X$, where
$X$ denotes any possible particle combination. The $\Sigma^-$ candidates are
tagged in $J/\psi \to \bar{\Sigma}(1385)^+\Sigma^-$ decays. No signal
candidates are found, and the upper limits on the branching fractions at the
90\% confidence level are determined to be $\mathcal{B}(\Sigma^{-} \to p e^{-}
e^{-}) < 6.7 \times 10^{-5}$ and $\mathcal{B}(\Sigma^{-} \to \Sigma^{+} X) <
1.2 \times 10^{-4}$.
