Let $L$ and $N$ be two Noetherian $R$-modules ($R$ is a commutative ring with 1). Is it right that $L \otimes_R N$ is Noetherian?
If not, what additional conditions on $L$ and $N$ are required in order that the statement will be true?
If every submodule of $L \otimes_R N$ is of the form $L_0 \otimes_R N_0$ where $L_0 \subset L$ and $N_0 \subset N$ are $R$-submodules, then each of them is finitely-generated and thus also their product and we are done. The question is if any submodule of the tensor product can be written as $L_0 \otimes_R N_0$?