Symmetric Derivations on Kahler Modules

In this paper, we define generalized symmetric derivations on K"{a}hler modules. We give the relationships between the projective dimensions of ${\mathrm{\Omega }}^{(1)}(R\mathrm{/}k)$ and ${\mathrm{\Omega }}^{(2)}(R\mathrm{/}k)$ 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.