I'm looking for an answer to these 2 trivially formulated problems:
Problem 1: Give an example of a ring $A$, an injective $A$-module homomorphism $f : M → M'$ and an $A$-module $N$ such that the homomorphism $\operatorname{id} ⊗ f : N ⊗_A M → N ⊗_A M'$ is not injective
Problem 2: Give an example of a module $M$ over a Noetherian ring $A$ such that $M_{\mathfrak p}$ is finitely generated for each $\mathfrak p ∈ \operatorname{Spec}(A)$, but $M$ is not finitely generated. $M_{\mathfrak p}$ is a localisation of M with multiplicatively closed subset being $A\setminus \mathfrak p$.
Last week I spent an hour thinking about it but did not get a result. This is not my homework so don't be afraid to help me. copies are already taken, that's why I ask here if I don't find answers in Google :) Thanks in advance.
For the first question, you can insider the canonical injection $i:I\hookrightarrow A$ of an ideal $I$ into the ring, and tensor by an $A$-module $N$ annihilated by $I$ (e.g. $A/I$: the image of the homomorphism $$i\otimes \operatorname{1}_N; I\otimes_A N\rightarrow A\otimes_AN\simeq N$$ is the submodule $IN=\{0\}$, so $i\otimes \operatorname{1}_N$ is not injective if $I\otimes_AN\ne\{0\}$, which happens for instance if $I$ is free and $N\ne \{0\}$.