SUBLATTICES OF R-TORS INDUCED BY A-MODULES
This article is concerned with the study of the sublattice gen (ρ) of R-tors, where ρ is some arbitrary but fixed member of R-tors. We use the concept of the ρ-A-module (M is a ρ-A-module, if M is ρ-torsion free and ρ ∨ξ ({M}) is an atom in gen (ρ)) and we define an equivalence relation in the sublattice gen (ρ). The partition associated to this equivalence relation allows us to get interesting information about this sublattice. As an application, we obtain new characterizations of ρ-artinian rings, ρ-semiartinian rings (a ring R is ρ-semiartinian if every non-zero ρ-torsion free R-module contains a τ- cocritical submodule) and rings with ρ-atomic dimension (rings such that for all σ ∈ gen (ρ) with σ 6= χ, there exists a σ-A-module).
Keywords:
torsion theories, lattices, A-modules, atomic dimension, Gabriel dimension left τ-semiartinian rings, left τ-artinian rings,
___
- G. Aguilar and F. Raggi, Sublattices of R-tors induced by the skeleton, Comm. Algebra, 21(4) (1993), 1347-1358.
- E. Barbut and W. Brandal, Localizations of torsion theories, Pacific J. Math., (1) (1983)27-37.
- J. Bueso and P. Jara, Semiartinian modules relative to a torsion theory, Comm. Algebra, 13(3) (1985), 631-644.
- J. Castro, J. R´ıos, and M. Teply, Torsion theoretic dimensions and relative Gabriel correspondence, J. Pure Appl. Algebra, 178 (1) (2003), 101-114.
- J. Castro, F. Raggi, J. R´ıos and J. Van den Berg, On the atomic dimension in module categories, Comm. Algebra, 33 (2005), 4679-4692.
- J. Castro, F. Raggi, and J. R´ıos, Decisive dimension and other related torsion theoretic dimensions, J. Pure Appl. Algebra, 209 (2007), 139-149
- J. Golan, Torsion Theories, Longman Scientific & Technical, Harlow, 1986.
- J. Golan, On the Cantor-Bendixon-Simmons filtration of a torsion theory, Comm. Algebra, 16(4) (1988), 681-688.
- J. Golan and H. Simmons, Derivatives, Nuclei and Dimensions on the Frame of Torsion Theories, Longman Scientific and Technical , Harlow, 1988.
- W.J. Lewis, The spectrum of a ring as a partially ordered set, J. Algebra, 25 (1973), 419-434.
- R. Miller and M. Teply, The descending chain condition relative to a torsion theory, Pacific J. Math, 83 (1979), 207-219.
- C. Nastasescu, Modules injectifs de type fini par rapport a une topologie addi- tive, Comm. Algebra, 9 (1981), 67-79
- B. Stenstr¨om, Rings of Quotients, Die Grundlehren der Math. Wiss. in Eizeld, Vol 217, Springer, Berlin, 1975.
- A.M. Viola-Prioli and J. E. Viola-Prioli, Rings whose kernel functors are lin- early ordered, Pacific J. Math., 132(1) (1988), 21-34. Jaime Castro P´erez
- Departamento de Matem´aticas Instituto Tecnol´ogico y de Estudios Superiores de Monterrey Calle del Puente 222, Tlalpan M´exico, D.F. M´exico e-mail: jcastrop@itesm.mx Gerardo Aguilar S´anchez Departamento de F´ısica y Matem´aticas Divisi´on de Ingenier´ıa y Arquitectura Tecnol´ogico de Monterrey Campus Ciudad de M´exico M´exico e-mail: gerardo.aguilar@itesm.mx
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI