Let $k$ be a field of characteristic $p$ and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a $k$-algebra $A$ (non commutative) and an $A\otimes_kk(x)$-module $M$, that is finite dimensional. If $M$ is indecomposable and not defined over $k$ (i.e. there does not exist an $A$-module $N$ such that $M=N\otimes_kk(x)$ is it true that $M$ is still indecomposable, when viewed as an $A$-module?
This question arised when considering the case of the map $k(x)\to k(x)$ given by multiplication by $x$. We may view this as a module under $k[T]$. If you want, assume that $A$ is finite dimensional (even if the original case would not really be covered).