I need to prove this theorem, but I guess I need some hints since I cannot even imagine how it can be proved. By the way, it is known that if $R$ is isonoetherian, then it is nei-Noetherian. The followings are the definitions of some terms in the theorem:
Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called $\textbf{isonoetherian}$ if, for every ascending chain $M_1 \leq M_2 \leq \cdots$ of submodules of $M$, it can be found an index $n$ $\geq1$ such that $M_n \cong M_i$ for every $i$ $\geq$ $n$. If $R_R$ is isonoetherian, then $R$ is a right isonoetherian ring.
An $R$-module $M$ is called $\textbf{nei-Noetherian}$ if for all ascending chains $M_1 \leq M_2 \leq \cdots$ of non-essential submodules of M, there can be found an index $n$ such that $M_i \cong M_n$, for each $i \geq n$.