I'm currently reading Sarah Witherspoon's book on Hochschild Cohomology. At the beginning of the fourth chapter it is given the following definition:
Definition 1. If $k$ is a field and $A$ is a $k$-algebra, we say that $A$ is smooth if its Hochschild dimension is finite and it has a finite projective resolution as an $A^e$-module by finitely generated projective modules.
However, in Weibel's book, the definition for smoothness is different (and it is given only for commutative $k$-algebras):
Definition 2. A commutative $k$-algebra $A$ is called smooth if for every square-zero extenstion $0\rightarrow M \rightarrow E \xrightarrow{\varepsilon} T \rightarrow 0$ of commuatative $k$-algebras and every algebra map $\nu:A\rightarrow T$, there is a $k$-algebra map $u:A\rightarrow E$ such that $\varepsilon u = \nu$.
As Witherspoon indicate in her book at page 51, these definitions should match for finitely generated commutative $k$-algebra $A$. Why is this so?
