Solution
= Solution
The statement is true. If $M$ is generated by $m_1,\ldots,m_r$, there is a surjection $R^r\twoheadrightarrow M$. The right exactness of the <tensor product of modules> gives a surjection
$$
N^r\cong R^r\otimes_RN\twoheadrightarrow M\otimes_RN.
$$
A finite direct sum of <Noetherian modules> is Noetherian, and a quotient of a Noetherian module is Noetherian. Hence $M\otimes_RN$ is Noetherian.