Baer duality for commutative rings

Pham Ngoc Ánh; Dolors Herbera; Claudia Menini

doi:https://doi.org/10.1515/form.2002.005

Baer duality for commutative rings

Pham Ngoc Ánh, Dolors Herbera, Claudia Menini

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

Abstract

A Baer duality is a triple (R, RUT, T), consisting of rings R, T and a bimodule RUT faithful on both sides, such that the lattices of submodules ℒ(RR) and ℒ(UT), as well as ℒ(RU) and ℒ(TT), are anti-isomorphic. The theory of Baer duality for commutative rings is developed. Analogously to the Morita duality case, it is shown that any commutative ring with Baer duality has self-duality. The existence of the lattice anti-isomorphisms in a Baer duality implies that all the lattices involved satisfy Grothendieck's condition AB5. It is showed that any AB5 commutative domain has Baer duality. An example of an AB5 commutative ring without Baer duality is given.

Original language	English
Pages (from-to)	47-63
Journal	Forum Mathematicum
Volume	14
DOIs	https://doi.org/10.1515/form.2002.005
Publication status	Published - 1 Jan 2002

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title = "Baer duality for commutative rings",

abstract = "A Baer duality is a triple (R, RUT, T), consisting of rings R, T and a bimodule RUT faithful on both sides, such that the lattices of submodules ℒ(RR) and ℒ(UT), as well as ℒ(RU) and ℒ(TT), are anti-isomorphic. The theory of Baer duality for commutative rings is developed. Analogously to the Morita duality case, it is shown that any commutative ring with Baer duality has self-duality. The existence of the lattice anti-isomorphisms in a Baer duality implies that all the lattices involved satisfy Grothendieck's condition AB5. It is showed that any AB5 commutative domain has Baer duality. An example of an AB5 commutative ring without Baer duality is given.",

author = "{\'A}nh, {Pham Ngoc} and Dolors Herbera and Claudia Menini",

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T1 - Baer duality for commutative rings

AU - Ánh, Pham Ngoc

AU - Herbera, Dolors

AU - Menini, Claudia

PY - 2002/1/1

Y1 - 2002/1/1

N2 - A Baer duality is a triple (R, RUT, T), consisting of rings R, T and a bimodule RUT faithful on both sides, such that the lattices of submodules ℒ(RR) and ℒ(UT), as well as ℒ(RU) and ℒ(TT), are anti-isomorphic. The theory of Baer duality for commutative rings is developed. Analogously to the Morita duality case, it is shown that any commutative ring with Baer duality has self-duality. The existence of the lattice anti-isomorphisms in a Baer duality implies that all the lattices involved satisfy Grothendieck's condition AB5. It is showed that any AB5 commutative domain has Baer duality. An example of an AB5 commutative ring without Baer duality is given.

AB - A Baer duality is a triple (R, RUT, T), consisting of rings R, T and a bimodule RUT faithful on both sides, such that the lattices of submodules ℒ(RR) and ℒ(UT), as well as ℒ(RU) and ℒ(TT), are anti-isomorphic. The theory of Baer duality for commutative rings is developed. Analogously to the Morita duality case, it is shown that any commutative ring with Baer duality has self-duality. The existence of the lattice anti-isomorphisms in a Baer duality implies that all the lattices involved satisfy Grothendieck's condition AB5. It is showed that any AB5 commutative domain has Baer duality. An example of an AB5 commutative ring without Baer duality is given.

U2 - https://doi.org/10.1515/form.2002.005

DO - https://doi.org/10.1515/form.2002.005

M3 - Article

VL - 14

SP - 47

EP - 63

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

ER -