On a book I’m studying I found this statement: given $I$ an ideal of $K[X]$ (polynomial ring in n variables) then:
$ Der_k(A) = \{ D \in Der_k(K[X]) \; | \; D(I) \subset I \} $
Where $A = K[X]/I$. I feel this is not true: given the fact that $K[X]$ has a free module of kahler differentials I can lift every derivation of $A$, but there is no reason for which this lifting should be unique. Am I right? If not please give the proof of the statement. Thank you.