What are the submodules of $M⊗K$?

Question

Let $k$ be a field and $K$ is a extension of this field.

And let $A$ be a finite type $k$-algebra, and $M$ be a finitely generated module over $A$.

Then, is the form of submodules of $M⊗_kK$ always $N⊗_kK$? (where $N$ is an $A$-submodule of $M$)

Answer 1

No. Small examples suffice: take $k=\mathbb{F}_p$, the finite field of $p$ elements, and take $K=\mathbb{F}_{p^2}$. Now, take $A=k$ and $M=k^2$. Then $M$ only has $p+1$ non-trivial submodules, while $M\otimes_k K$ (which is isomorphic to $K^2$) has $p^2+1$ non-trivial submodules. Thus $M\otimes_k K$ must have submodules which are not of the form $N\otimes_k K$, with $N$ a submodule of $M$.