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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: Suppose $ R\xrightarrow{\varphi }S$ 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

$\displaystyle S{ \otimes _R}S\xrightarrow{{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow{\varphi }S,$

which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring.

Suppose R is an overing of B. Let $ {C_B} = R{ \otimes _B}R$. There are maps

\begin{displaymath}\begin{array}{*{20}{c}} {{C_B} = R{ \otimes _B}R\xrightarrow{... ...{r_1} \otimes {r_2} \to {r_1}{r_2})}}R.} \hfill \\ \end{array} \end{displaymath}

These maps give $ {C_B}$ an R-coring structure. The dual $ ^\ast{C_B}$ is naturally isomorphic to the ring $ {\text{End}_{{B^ - }}}R$ 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 $ {C_{\text{Z}}}$. Then $ ^\ast{C_{\text{Z}}}$ is $ {\text{End}_{\text{Z}}}R$, 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 $ {\text{End}_{\text{Z}}}R$, both major ingredients of the Jacobson-Bourbaki theorem.

$ 1 \otimes 1$ is a ``grouplike'' element in the R-coring $ {C_{\text{Z}}}$ (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 $ {C_{\text{Z}}} \to {C_B}$ is a surjective coring map. Conversely if $ {C_{\text{Z}}}\xrightarrow{\pi }D$ is a (surjective) coring map then $ \pi (1 \otimes 1)$ is a grouplike in D and $ \{ r \in R\vert r\pi (1 \otimes 1) = \pi (1 \otimes 1)r\} $ is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of $ C{ \otimes _{\text{Z}}}C$ 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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Keywords: Coring, coalgebra, division ring
Article copyright: © Copyright 1975 American Mathematical Society

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