The separable closure of some commutative rings

Magid, Andy R.

doi:10.1090/S0002-9947-1972-0311642-3

The separable closure of some commutative rings
HTML articles powered by AMS MathViewer

by Andy R. Magid PDF

Trans. Amer. Math. Soc. 170 (1972), 109-124 Request permission

Abstract:

The separable closure of a commutative ring with an arbitrary number of idempotents is defined and its Galois theory studied. Projective separable algebras over the ring are shown to be determined by the ’Galois groupoid’ of the closure. The existence of the closure is demonstrated for certain rings.

References

Lectures on topics in algebraic $K$-theory

Frank DeMeyer and Edward Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971. MR 0280479
Pierre Gabriel, Construction de préschémas quotient, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963/64) Inst. Hautes Études Sci., Paris, 1963, pp. Fasc. 2a, Exposé 5, 37 (French). MR 0257095
G. J. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461–479. MR 210699, DOI 10.1090/S0002-9947-1966-0210699-5
A. R. Magid, Galois groupoids, J. Algebra 18 (1971), 89–102. MR 272775, DOI 10.1016/0021-8693(71)90128-1
Andy R. Magid, Locally Galois algebras, Pacific J. Math. 33 (1970), 707–724. MR 263805
A. R. Magid, Pierce’s representation and separable algebras, Illinois J. Math. 15 (1971), 114–121. MR 272832
O. E. Villamayor and D. Zelinsky, Galois theory for rings with finitely many idempotents, Nagoya Math. J. 27 (1966), 721–731. MR 206055
O. E. Villamayor and D. Zelinsky, Galois theory with infinitely many idempotents, Nagoya Math. J. 35 (1969), 83–98. MR 244238
Shizuo Endo and Yutaka Watanabe, On separable algebras over a commutative ring, Osaka Math. J. 4 (1967), 233–242. MR 227211
R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR 0217056
Albert Wilansky, Topology for analysis, Ginn (A Xerox company), Waltham, Mass.-Toronto, Ont.-London, 1970. MR 0451181
Frank DeMeyer, Separable polynomials over a commutative ring, Rocky Mountain J. Math. 2 (1972), no. 2, 299–310. MR 294321, DOI 10.1216/RMJ-1972-2-2-299
Richard F. Arens and Irving Kaplansky, Topological representation of algebras, Trans. Amer. Math. Soc. 63 (1948), 457–481. MR 25453, DOI 10.1090/S0002-9947-1948-0025453-6
S. U. Chase, D. K. Harrison, and Alex Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33. MR 195922