Strongly separable pairings of rings

Cunningham, Robert S.

doi:10.1090/S0002-9947-1970-0255615-6

Strongly separable pairings of rings
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by Robert S. Cunningham

Trans. Amer. Math. Soc. 148 (1970), 399-416

DOI: https://doi.org/10.1090/S0002-9947-1970-0255615-6

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Abstract:

The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Müller, Nakayama, and others. We use adjoint functors to define a pairing of the two rings and develop a theory of relative projective and injective modules for pairings generalizing that of Hochschild for extensions. The main purpose of this paper is to define “strongly separable pairings” generalizing strongly separable (i.e. finitely generated projective separable) algebras. We show that such pairings have very close connections to category equivalences, so that it is natural to investigate those properties shared by two rings which admit a strongly separable pairing. We show that most “categorical” properties are so shared.

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The Morita theorems

Algèbre commutative

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