nLab smooth algebra in algebraic sense

In $n$lab, the entry smooth algebra is dedicated to the notion of $C^\infty$-rings, hence it is coming from an extension of the theory of differentiable manifolds. On the other hand in commutative algebra and in the study of associative algebras there are many notions of smoothness, regularity and formal smoothness which correspond to nonsingularity in an algebraic category, while the algebras are quite different from the $C^\infty$-setup. In algebraic geometry, Grothendieck introduced many notions of smoothness for commutative algebraic schemes (they say 19 different notions). See also formally smooth morphism.

• W.F. Schelter, Smooth algebras, J. Algebra 103 (1986), 677–685.