  Motivated by deformation quantization, we introduced in an earlier work the
notion of formal Morita equivalence in the category of $^*$-algebras over a
ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring
$\ring R$. The goal of the present paper is twofold. First, we clarify the
relationship between formal Morita equivalence, Ara's notion of Morita
$^*$-equivalence of rings with involution, and strong Morita equivalence of
$C^*$-algebras. Second, in the general setting of $^*$-algebras over $\ring C$,
we define `closed' $^*$-ideals as the ones occuring as kernels of
$^*$-representations of these algebras on pre-Hilbert spaces. These ideals form
a lattice which we show is invariant under formal Morita equivalence. This
result, when applied to Pedersen ideals of $C^*$-algebras, recovers the
so-called Rieffel correspondence theorem. The triviality of the minimal element
in the lattice of closed ideals, called the `minimal ideal', is also a formal
Morita invariant and this fact can be used to describe a large class of
examples of $^*$-algebras over $\ring C$ with equivalent representation theory
but which are not formally Morita equivalent. We finally compute the closed
$^*$-ideals of some $^*$-algebras arising in differential geometry.
