A module is said to be hollow if, its every proper submodule is small, that is, for any two proper submodule $N,K$ of $M$, $N+K\neq M$. A module is said to be Artinian if it satisfies descending chain condition on its submodules.
I am seeking for an example which is hollow but not Artinian.
Let $K$ be a field and let $V$ be a $K$-vector space. The ring $R=K\oplus V$, where elements $(k_1,v_1)$ and $(k_2,v_2)$ are added and multiplied by the rules $(k_1,v_1)+(k_2,v_2)=(k_1+k_2,v_1+v_2)$ and $(k_1,v_1)\cdot(k_2,v_2)=(k_1k_2,k_1v_2+k_2v_1)$, is a commutative local ring with maximal ideal $0\oplus V$. If you consider $R$ as an $R$-module, written ${}_RR$, then its submodule lattice is the ideal lattice of the ring $R$, which is isomorphic to the subspace lattice of $V$ with a new top element adjoined. So ${}_RR$ is hollow. It is Artinian only if $V$ is finite dimensional.