  In this paper, we define generalized symmetric derivations on K\"{a}hler
modules. We give the relationships between the projective dimensions of
$\Omega^{(1)}(R/k)$ and $\Omega^{(2)}(R/k)$ by using the symmetric
derivation.We then give some interesting results by using this definition and
related to K\"{a}hler modules and symmetric derivations.
