Let $X$ be a smooth projective variety over a field, say $k$. Consider the short exact sequence of $k$-modules, $$0 \to A_1 \to A \to A_2 \to 0$$ where $A$ and $A_2$ are $k$-algebras. Since these can also be seen as $k$-vector spaces, the short exact sequence splits. Hence, we can write $A \cong A_2 \oplus A_1$. This should mean for any $\mathcal{O}_X$-module, $\mathcal{F}$ we have $A \otimes \mathcal{F} \cong (A_1 \otimes \mathcal{F}) \oplus (A_2 \otimes \mathcal{F})$. Is this correct?
The motivation of the problem comes from (infinitesimal) deformation theory, where one can replace $A, A_2$ by local Artinian $k$-algebras such that $A_2$ is an infinitesimal extension of $A$ by a nilpotent ideal $A_1$.