The predual theorem to the Jacobson-Bourbaki theorem

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 \[ S{ \otimes _R}S\xrightarrow {{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow {\varphi }S,\] 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 \[ \begin {array}{*{20}{c}} {{C_B} = R{ \otimes _B}R\xrightarrow {{({r_1} \otimes {r_2} \to {r_1} \otimes 1 \otimes {r_2})}}R{ \otimes _B}R{ \otimes _B}R = ({C_B}){ \otimes _R}({C_B}),} \hfill \\ {{C_B} = R{ \otimes _B}R\xrightarrow {{({r_1} \otimes {r_2} \to {r_1}{r_2})}}R.} \hfill \\ \end {array} \] 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 $\{ r \in R|r\pi (1 \otimes 1) = \pi (1 \otimes 1)r\}$ 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.