Étale morphism of rings

Remarks

Condition 2 says that the lift required by formal smoothness exists and is unique. Equivalent forms: $A$ is smooth and unramified over $R$; or $A$ is finitely presented, flat and unramified over $R$.

Example: let $f \in R[x]$ be monic and let $A = R[x]/(f)$; if the class of $f'$ is a unit in $A$, then $A$ is étale over $R$. For instance, $R[x]/(x^2 - a)$ is étale when $2a$ is a unit of $R$, since $(2x)^2 = 4a$ in $A$. Over a field $k$, a finite separable extension $L$ of $k$ is étale, and the étale $k$-algebras are exactly the finite products of such fields.

Non-example: $A = k[x]$ over a field $k$ is smooth and finitely presented, but a map $A \to C/J$ has many lifts: $x$ may go to any preimage of $u(x)$. So uniqueness in clause 2 fails; accordingly $\Omega_{A/k} = A \, dx \ne 0$, see Kähler differentials.