A (commutative) algebra $A$ is called formally smooth if for any (commutative) algebra $R$ and an ideal $I\subset R$ such that $I^2=0$, any morphism $A\to R/I$ lifts to a morphism $A\to R$.
Suppose now that $X$ is a variety and $A$ is the algebra of regular functions on it.
How are the definitions of formal smoothness of $A$ and smoothness of the variety $X$ related?
Are the two notions equivalent? It doesn't seem to be the case, but I don't know how to prove it. I might be very silly here. In this case I would be glad if you explain that to me.
Thank you very much!