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The main goal of this paper is to characterize the module of K\"ahlerdifferentials for an extension of valuation rings. More precisely, we considera simple algebraic valued field extension $(L/K,v)$ and the correspondingvaluation rings $\VR_L$ and $\VR_K$. In the case when $e(L/K,v)=1$ we present acharacterization for $\Omega_{\VR_L/\VR_K}$ in terms of a given sequence of keypolynomials for the extension. Moreover, we use our main result to present acharacterization for when $\Omega_{\VR_L/\VR_K}=\{0\}$.

The module of K\"ahler differentials for extensions of valuation rings