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
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
October 2010
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), .
(doi:10.1007/s00012-010-0092-1).
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.
This record has no associated files available for download.
More information
Published date: October 2010
Keywords:
Gödel algebras, Heyting algebras, Monoid of ideals, Noncommutative rings, Von Neumann regular rings
Identifiers
Local EPrints ID: 416723
URI: http://eprints.soton.ac.uk/id/eprint/416723
ISSN: 0002-5240
PURE UUID: 2a9f2dbf-fb87-498e-b8b3-c8d1d1b29ae5
Catalogue record
Date deposited: 05 Jan 2018 17:30
Last modified: 15 Mar 2024 17:37
Export record
Altmetrics
Contributors
Author:
L.P. Belluce
Author:
Antonio Di Di Nola
Author:
Enrico Marchioni
Download statistics
Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.
View more statistics