Let $k$ be a field, and let $A$ be a commutative $k$-algebra.
Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal.
Consider the ideal $I\otimes_k A \subseteq A\otimes_k A$. Is this ideal, considered as an $A\otimes_k A$-module, a noetherian module? is it coherent?
The answer is typically no. E.g. if $I$ is a principal ideal, and say $A$ is a domain, then $I$ is isomorphic to $A$ as an $A$-module, so $I\otimes_kA$ is isomorphic to $A\otimes_k A$ as an $A\otimes_k A$-module. Typically $A\otimes_k A$ will not be Noetherian (and I don't think it will be coherent in general either, although I've not thought much about that, so don't have a counterexample at hand).