On lattices associated to rings with respect to a preradical

Yıl 2022, , 13 - 37, 17.01.2022

Erwin Cerda-leon Hugo Rıncon-mejıa

https://doi.org/10.24330/ieja.1058385

Öz

We introduce some new lattices of classes of modules with respect to appropriate preradicals. We introduce some concepts associated with these
lattices, such as the $\sigma$-semiartinian rings, the $\sigma$-retractable modules, the $\sigma$-$V$-rings, the $\sigma$-max rings. We continue to study $\sigma$-torsion theories, $\sigma$-open classes, $\sigma$-stable classes. We prove some theorems that extend some known results. Our results fall into well known situations when the preradical $\sigma$ is chosen as the identity preradical.

Anahtar Kelimeler

Preradical, lattice of module class, hereditary torsion class, boolean lattice, V-ring, max-ring

Kaynakça

• A. Alvarado-Garcia, H. Rincon-Mejia and J. Rios-Montes, On the lattices of natural and conatural classes in R-Mod, Comm. Algebra, 29(2) (2001), 541-556.
• A. Alvarado-Garcia, H. Rincon-Mejia and J. Rios-Montes, On some lattices of module classes, J. Algebra Appl., 5 (1) (2006), 105-117.
• A. Alvarado-Garcia, H. Rincon-Mejia, J. Rios-Montes and B. Tome-Arreola, On conatural classes and cotype submodules, Int. Electron. J. Algebra, 11 (2012), 64-81.
• A. Alvarado-Garcia, C. Cejudo-Castilla, H. Rincon-Mejia, F. Vilchis-Montalvo and M. Zorrilla-Noriega, On boolean lattices of module classes, Algebra Colloq., 25(2) (2018), 285-294.
• L. Bican, T. Kepka and P. Nemec, Rings, Modules and Preradicals, Marcel Dekker, Inc., New York, 1982.
• E. Cerda-Leon and H. Rincon-Mejia, Big lattices of module classes induced by preradicals, Sao Paulo J. Math. Sci., 14 (2020), 185-206.
• J. Dauns and Y. Zhou, Classes of Modules, Chapman & Hall/CRC, Boca Raton, FL, 2006.
• S. E. Dickson, A torsion theory for abelian categories, Trans. Amer. Math. Soc., 121 (1966), 223-235.
• J. Golan, Torsion Theories, Longman Scientific & Technical, New York, 1986.
• T. Kosan and M. J. Zemlicka, Mod-retractable rings, Comm. Algebra, 42 (2014), 998-1010.
• F. Raggi, J. Rios, H. Rincon-Mejia, R. Fernandez-Alonso and C. Signoret, The lattice structure of preradicals, Comm. Algebra, 30(3) (2002), 1533-1544.
• H. Rincon-Mejia and M. Zorrilla-Noriega, On some relations between the lattices R-nat, R-conat and R-tors and the rings they characterize, J. Algebra Appl., 12(5) (2013), 1250214, 13 pp.
• P. F. Smith, Modules with many homomorphisms, J. Pure Appl. Algebra, 197 (2005), 305-321.
• B. Stenstrom, Rings of Quotients, Springer-Verlag, New York Heidelberg, 1975.