Smooth algebras
The problem underlying this paper is the search for an appropriate analogue, in the setting of prime P.I. algebras over an algebraically closed field k, of the notion of a nonsingular variety. Such an algebra A is here called smooth if, whenever A occurs as the factor of a k-algebra C by an ideal N of square zero, there is a copy of A in C splitting the extension of N by A. Moreover, C should only be allowed to come from a category of algebras satisfying the same identities as A. The main result proved here uses [\textit{A. Brumer}, Bull. Am. Math. Soc. 69, 721-724 (1963); addendum 70, 185 (1964; Zbl 0113.260)] to demonstrate that this is a sensible definition in dimension 1, by showing that if A is an algebra as above, and A has Krull dimension 1, then A is smooth (in the category of associative k-algebras) if and only if it is smooth in the category of k-algebras satisfying the same identities as A if and only if all right ideals of A are projective if and only if it is geometrically integrally closed. The last concept here is defined in [\textit{M. Artin} and \textit{W. Schelter}, Adv. Math. 39, 289-329 (1981; Zbl 0461.16014)].
scientific article; zbMATH DE number 3975143