On cycles in intersection graphs of rings

‎Let $R$ be a commutative ring with non-zero identity‎. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph < s > of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring‎. ‎Also‎, ‎we shall describe all complete‎, ‎regular and $n$-claw-free intersection graphs‎. ‎Finally‎, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs‎. ‎We show that such graphs are indeed pancyclic.