The predual theorem to the Jacobson-Bourbaki theorem

Author:
Moss Sweedler

Journal:
Trans. Amer. Math. Soc. **213** (1975), 391-406

MSC:
Primary 16A49; Secondary 16A24, 16A74

MathSciNet review:
0387345

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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

*R-coring*.

Suppose *R* is an overing of *B*. Let . There are maps

*R*-coring structure. The dual is naturally isomorphic to the ring of

*B*-linear endomorphisms of

*R*considered as a left

*B*-module. 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

*R*-corings correspond to subrings of

*R*and subrings of , both major ingredients of the Jacobson-Bourbaki theorem.

is a ``grouplike'' element in the *R*-coring (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 Jacobson-Bourbaki correspondence is dual to the above correspondence.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0387345-9

Keywords:
Coring,
coalgebra,
division ring

Article copyright:
© Copyright 1975
American Mathematical Society