Multiplicative residue semigroups with planar Cayley graphs

We study multiplicative residue semigroups that admit planar Cayley graphs. It is proved that the multiplicative semigroup of the residue ring $\mathbb{Z}_n$ admits a planar Cayley graph if and only if $n=4,6,8$ or $n$ is a prime number. Examples of minimal systems of generators of multiplicative residue semigroups with respect to some modules and their Cayley graphs are given, illustrating the obtained results.