Smooth algebra

In algebra, a commutative "k"-algebra "A" is said to be 0-smooth if it satisfies the following lifting property: given a "k"-algebra "C", an ideal "N" of "C" whose square is zero and a "k"-algebra map formula_1, there exists a "k"-algebra map formula_2 such that "u" is "v" followed by the canonical map. If there exists at most one such lifting "v", then "A" is said to be 0-unramified (or 0-neat). "A" is said to be 0-étale if it is 0-smooth and 0-unramified. The notion of 0-smoothness is also called formal smoothness.
A finitely generated "k"-algebra "A" is 0-smooth over "k" if and only if Spec "A" is a smooth scheme over "k".
A separable algebraic field extension "L" of "k" is 0-étale over "k". The formal power series ring formula_3 is 0-smooth only when formula_4 and formula_5 (i.e., "k" has a finite "p"-basis.)
"I"-smooth.
Let "B" be an "A"-algebra and suppose "B" is given the "I"-adic topology, "I" an ideal of "B". We say "B" is I"-smooth over "A if it satisfies the lifting property: given an "A"-algebra "C", an ideal "N" of "C" whose square is zero and an "A"-algebra map formula_6 that is continuous when formula_7 is given the discrete topology, there exists an "A"-algebra map formula_8 such that "u" is "v" followed by the canonical map. As before, if there exists at most one such lift "v", then "B" is said to be I"-unramified over "A (or I"-neat). "B" is said to be I"-étale if it is I"-smooth and I"-unramified. If "I" is the zero ideal and "A" is a field, these notions coincide with 0-smooth etc. as defined above.
A standard example is this: let "A" be a ring, formula_9 and formula_10 Then "B" is "I"-smooth over "A".
Let "A" be a noetherian local "k"-algebra with maximal ideal formula_11. Then "A" is formula_11-smooth over "k" if and only if formula_13 is a regular ring for any finite extension field formula_14 of "k".