The predual theorem to the Jacobson-Bourbaki theorem
Authors: Sweedler Moss 
Published: July 2010

Suppose

R

→

φ

S

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S circled-times Subscript upper R Baseline upper S right-arrow Overscript left-parenthesis s 1 circled-times s 2 right-arrow s 1 s 2 right-parenthesis Endscripts upper S comma upper R right-arrow Overscript phi Endscripts upper S comma">

S

⊗
R

S

→

(

s
1

⊗

s
2

→

s
1

s
2

)

S
,

R

→

φ

S
,

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

⊗
B

R

{C_B} = R{ \otimes _B}R

. There are maps
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper C Subscript upper B Baseline equals upper R circled-times Subscript upper B Baseline upper R right-arrow Overscript left-parenthesis r 1 circled-times r 2 right-arrow r 1 circled-times 1 circled-times r 2 right-parenthesis Endscripts upper R circled-times Subscript upper B Baseline upper R circled-times Subscript upper B Baseline upper R equals left-parenthesis upper C Subscript upper B Baseline right-parenthesis circled-times Subscript upper R Baseline left-parenthesis upper C Subscript upper B Baseline right-parenthesis comma 2nd Row upper C Subscript upper B Baseline equals upper R circled-times Subscript upper B Baseline upper R right-arrow Overscript left-parenthesis r 1 circled-times r 2 right-arrow r 1 r 2 right-parenthesis Endscripts upper R period EndLayout">

C
B

=
R

⊗
B

R

→

(

r
1

⊗

r
2

→

r
1

⊗
1
⊗

r
2

)

R

⊗
B

R

⊗
B

R
=
(

C
B

)

⊗
R

(

C
B

)
,

C
B

=
R

⊗
B

R

→

(

r
1

⊗

r
2

→

r
1

r
2

)

R
.

\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

{C_B}

an R-coring structure. The dual

∗

C
B

^\ast {C_B}

is naturally isomorphic to the ring

End

B
−

R

{\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

Z

{C_{\text {Z}}}

. Then

∗

C

Z

^\ast {C_{\text {Z}}}

is

End

Z

R

{\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

End

Z

R

{\text {End}_{\text {Z}}}R

, both major ingredients of the Jacobson-Bourbaki theorem.

1
⊗
1

1 \otimes 1

is a “grouplike” element in the R-coring

C

Z

{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

Z

→

C
B

{C_{\text {Z}}} \to {C_B}

is a surjective coring map. Conversely if

C

Z

→

π

D

{C_{\text {Z}}}\xrightarrow {\pi }D

is a (surjective) coring map then

π
(
1
⊗
1
)

\pi (1 \otimes 1)

is a grouplike in D and

{
r
∈
R

|

r
π
(
1
⊗
1
)
=
π
(
1
⊗
1
)
r
}

\{ 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

⊗

Z

C

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.