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?