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Rings and Gödel algebras

Rings and Gödel algebras
Rings and Gödel algebras
In this paper, we study those rings whose semiring of ideals can be given the structure of a Gödel algebra. Such rings are called Gödel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Gödel ring R is a subdirect product of prime Gödel rings Ri, and the Gödel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into ∏i Id(Ri), where each Id(Ri) is the algebraic lattice of ideals of Ri that can be equipped with the structure of a Gödel algebra. We see that the mapping associating to each Gödel ring its Gödel algebra of ideals is functorial from the category of Gödel rings with epimorphisms into the full subcategory of frames whose objects are Gödel algebras and whose morphisms are complete epimorphisms. © 2010 Springer Basel AG.
Gödel algebras, Heyting algebras, Monoid of ideals, Noncommutative rings, Von Neumann regular rings
0002-5240
103-116
Belluce, L.P.
b64528ef-dab3-475a-849b-00743a35bf8a
Di Nola, Antonio Di
8516c1cb-0ebe-4613-ae2f-880fe43f5770
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96
Belluce, L.P.
b64528ef-dab3-475a-849b-00743a35bf8a
Di Nola, Antonio Di
8516c1cb-0ebe-4613-ae2f-880fe43f5770
Marchioni, Enrico
729c9984-5949-438e-8de7-0e079bdb9f96

Belluce, L.P., Di Nola, Antonio Di and Marchioni, Enrico (2010) Rings and Gödel algebras. Algebra Universalis, 64 (1), 103-116. (doi:10.1007/s00012-010-0092-1).

Record type: Article

Abstract

In this paper, we study those rings whose semiring of ideals can be given the structure of a Gödel algebra. Such rings are called Gödel rings. We investigate such structures both from an algebraic and a topological point of view. Our main result states that every Gödel ring R is a subdirect product of prime Gödel rings Ri, and the Gödel algebra Id(R) associated to R is subdirectly embeddable as an algebraic lattice into ∏i Id(Ri), where each Id(Ri) is the algebraic lattice of ideals of Ri that can be equipped with the structure of a Gödel algebra. We see that the mapping associating to each Gödel ring its Gödel algebra of ideals is functorial from the category of Gödel rings with epimorphisms into the full subcategory of frames whose objects are Gödel algebras and whose morphisms are complete epimorphisms. © 2010 Springer Basel AG.

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More information

Published date: October 2010
Keywords: Gödel algebras, Heyting algebras, Monoid of ideals, Noncommutative rings, Von Neumann regular rings

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Local EPrints ID: 416723
URI: http://eprints.soton.ac.uk/id/eprint/416723
ISSN: 0002-5240
PURE UUID: 2a9f2dbf-fb87-498e-b8b3-c8d1d1b29ae5

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Date deposited: 05 Jan 2018 17:30
Last modified: 13 Mar 2019 19:04

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Contributors

Author: L.P. Belluce
Author: Antonio Di Di Nola
Author: Enrico Marchioni

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