The predual theorem to the JacobsonBourbaki theorem
Author:
Moss Sweedler
Journal:
Trans. Amer. Math. Soc. 213 (1975), 391406
MSC:
Primary 16A49; Secondary 16A24, 16A74
MathSciNet review:
0387345
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Suppose is a map of rings. S need not be an R algebra since R may not be commutative. Even if R is commutative it may not have central image in S. Nevertheless the ring structure on S can be expressed in terms of two maps which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an Rcoring. Suppose R is an overing of B. Let . There are maps These maps give an Rcoring structure. The dual is naturally isomorphic to the ring of Blinear endomorphisms of R considered as a left Bmodule. In case B happens to be the subring of R generated by 1, we write . Then is , the endomorphism ring of R considered as an additive group. This gives a clue how certain Rcorings correspond to subrings of R and subrings of , both major ingredients of the JacobsonBourbaki theorem. is a ``grouplike'' element in the Rcoring (and should be thought of as a generic automorphism of R). Suppose R is a division ring and B a subring which is a division ring. The natural map is a surjective coring map. Conversely if is a (surjective) coring map then is a grouplike in D and is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of and the subrings of R which are division rings. We show how the JacobsonBourbaki correspondence is dual to the above correspondence.
 [1]
Henri
Cartan, Théorie de Galois pour les corps non
commutatifs, Ann. Sci. École Norm. Sup. (3) 64
(1947), 59–77 (French). MR 0023237
(9,325f)
 [2]
J.
Dieudonné, Linearly compact spaces and double vector spaces
over sfields, Amer. J. Math. 73 (1951), 13–19.
MR
0038962 (12,476a)
 [3]
G.
Hochschild, Double vector spaces over division rings, Amer. J.
Math. 71 (1949), 443–460. MR 0029889
(10,676a)
 [4]
N.
Jacobson, An extension of Galois theory to nonnormal and
nonseparable fields, Amer. J. Math. 66 (1944),
1–29. MR
0010554 (6,35f)
 [5]
N.
Jacobson, A note on division rings, Amer. J. Math.
69 (1947), 27–36. MR 0020981
(9,4c)
 [6]
, Lectures in abstract algebra. Vol. 3: Theory of fields and Galois theory, Van Nostrand, Princeton, N. J., 1964. MR 30 #3087.
 [7]
Moss
E. Sweedler, Hopf algebras, Mathematics Lecture Note Series,
W. A. Benjamin, Inc., New York, 1969. MR 0252485
(40 #5705)
 [1]
 H. Cartan, Théorie de Galois pour les corps non commutatifs, Ann. Sci. École Norm. Sup. (3) 64 (1947), 5977. MR 9, 325. MR 0023237 (9:325f)
 [2]
 J. Dieudonné, Linearly compact spaces and double vector spaces over s fields, Amer. J. Math. 73 (1951), 1319. MR 12, 476. MR 0038962 (12:476a)
 [3]
 G. Hochschild, Double vector spaces over division rings, Amer. J. Math. 71 (1949), 443460. MR 10, 676. MR 0029889 (10:676a)
 [4]
 N. Jacobson, An extension of Galois theory to nonnormal and nonseparable fields, Amer. J. Math. 66 (1944), 129. MR 6, 35. MR 0010554 (6:35f)
 [5]
 , A note on division rings, Amer. J. Math. 69 (1947), 2736. MR 9, 4. MR 0020981 (9:4c)
 [6]
 , Lectures in abstract algebra. Vol. 3: Theory of fields and Galois theory, Van Nostrand, Princeton, N. J., 1964. MR 30 #3087.
 [7]
 M. E. Sweedler, Hopf algebras, Math. Lecture Note Series, Benjamin, New York, 1969. MR 40 #5705. MR 0252485 (40:5705)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
16A49,
16A24,
16A74
Retrieve articles in all journals
with MSC:
16A49,
16A24,
16A74
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197503873459
PII:
S 00029947(1975)03873459
Keywords:
Coring,
coalgebra,
division ring
Article copyright:
© Copyright 1975
American Mathematical Society
