  We construct a large new family of simple modules over Takiff
$\mathfrak{sl}_{2}$. We prove that the quotient of the universal enveloping
algebra of the Takiff Lie algebra for $\mathfrak{sl}_{2}$ by the ideal
generated by a non-trivial central character is a simple algebra. In the case
of the trivial central character, we show that the corresponding ideal is
primitive by explicitly constructing a simple module whose annihilator
coincides with that ideal. Together with the annihilators of simple
$\mathfrak{sl}_{2}$-modules, we expect that the above ideals exhaust all
primitive ideal.
