The predual theorem to the Jacobson-Bourbaki theorem

Abstract

Suppose <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R right-arrow Overscript phi Endscripts upper S"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mover> <mml:mo>→</mml:mo> <mml:mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mml:mi> φ </mml:mi> </mml:mpadded> </mml:mover> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R\xrightarrow {\varphi }S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a map of rings. <italic>S</italic> need not be an <italic>R</italic> algebra since <italic>R</italic> may not be commutative. Even if <italic>R</italic> is commutative it may not have central image in <italic>S</italic> . Nevertheless the ring structure on <italic>S</italic> can be expressed in terms of two maps <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S circled-times Subscript upper R Baseline upper S right-arrow Overscript left-parenthesis s 1 circled-times s 2 right-arrow s 1 s 2 right-parenthesis Endscripts upper S comma upper R right-arrow Overscript phi Endscripts upper S comma"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi>R</mml:mi> </mml:msub> </mml:mrow> <mml:mi>S</mml:mi> <mml:mover> <mml:mo>→</mml:mo> <mml:mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mpadded> </mml:mover> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>R</mml:mi> <mml:mover> <mml:mo>→</mml:mo> <mml:mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mml:mi> φ </mml:mi> </mml:mpadded> </mml:mover> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S{ \otimes _R}S\xrightarrow {{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow {\varphi }S,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an <italic>R-coring</italic> . Suppose <italic>R</italic> is an overing of <italic>B</italic> . Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper B Baseline equals upper R circled-times Subscript upper B Baseline upper R"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{C_B} = R{ \otimes _B}R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . There are maps <disp-formula content-type="math/mathml"> \[ <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"> <mml:semantics> <mml:mtable rowspacing="4pt" columnspacing="1em"> <mml:mtr> <mml:mtd columnalign="left"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:mi>R</mml:mi> <mml:mover> <mml:mo>→</mml:mo>