The Predual Theorem to the Jacobson-Bourbaki Theorem

Abstract

ABSTRACT. Suppose R-*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 mapswhich satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring.Suppose R is an overing of B. Let CB= R ®B R. There are maps(rj <Sr2 -rxr2) CB = R®BR ->R.These maps give CB an Ä-coring structure. The dual *CB is naturally isomorphic to the ring End"._R of ¿-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 Cz. Then *CZ is EndzÄ, the endomorphism ring of R considered as an additive group. This gives a clue how certain Ä-corings correspond to subrings of R and subrings of EndzR, both major ingredients of the Jacobson-Bourbaki theorem.1 ¡8 1 is a "grouplike" element in the R-coring C^ (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^ -* CB is a surjective coring map. Conversely if C^ -*D is a (surjective) coring map then jt(1 CS 1) is a

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“…Throughout this paper and unless otherwise stated, k denote a commutative ring (with unit), A, A ′ , A ′′ , and B denote associative and unitary algebras over k, and C, C ′ , C ′′ , and D denote corings over A, A ′ , A ′′ , and B, respectively. We recall from [34] that an A-coring consists of an A-bimodule C with two A-bimodule maps…”

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“…Corings and comodules. Let A be an associative and unitary algebra over a commutative ring (with unit) k. We recall from [19] that an A-coring C consists of an A-bimodule C with two A-bimodule maps…”