Let $A$ be a finite dimensional commutative unital algebra over $\mathbb{K}$ (of most interest to me in $\mathbb{K}=\mathbb{R}$). Let $V$ be a finitely generated $A$-module and let $m:A\otimes_{\mathbb{K}} V\rightarrow V$ be the action of $A$ on $V$.
Then we have an exact sequence of $A$-modules $0\rightarrow ker(m)\rightarrow A\otimes_{\mathbb{K}} V\rightarrow V\rightarrow 0$ given by inclusion and then $m$.
Does this sequence always split (i.e is $A\otimes_{\mathbb{K}} V\cong ker(m)\oplus V$ canonically)?
No.
Example: $A= k[\epsilon]$, with defining relation $\epsilon^2=0$, and $V=A/(\epsilon)\simeq k$, so that $\epsilon \cdot 1 =0$. Then the exact sequence of $A$-modules
$$ 0 \to (\epsilon) \to A \to V \to 0$$ does not split because $\epsilon$ acts trivially on the kernel and cokernel. (Here $A\otimes_k V \simeq A$ as $A$-modules.)