Abstract
In this paper we introduce the big lattices R-sext and R-qext consisting the former of classes of left R-modules closed under isomorphisms, submodules and extensions and the later of classes closed under homomorphic images and extensions, respectively. We work with these two big lattices and study the consequences of assuming that they are the same proper class. We also consider big lattices of R-modules defined by other closure properties.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Alvarado, H. Rincón and J. Ríos, On the lattices of natural and conatural classes in R-mod, Comm. Algebra. 29 (2) (2001) 541–556.
A. Alvarado, H. Rincón and J. Ríos, On Some Lattices of Module Classes, Journal of Algebra and its Applications. 2006, 105–117.
L. Bican, T. Kepka, P. Němec, Rings, modules, and preradicals. Lecture Notes in Pure and Applied Mathematics, 75. Marcel Dekker, Inc., New York, 1982.
J. Dauns, Y. Zhou, Classes of Modules. Pure and Applied Mathematics (Boca Raton), 281. Chapman & Hall/CRC, Boca Raton, FL., 2006.
Fernández-Alonso Rogelio, Raggi, Francisco, The lattice structure of nonhereditary torsion theories. Comm. Algebra 26 (1998), no. 6, 1851–1861.
Golan, Jonathan, Torsion theories. Pitman Monographs and Surveys in Pure and Applied Mathematics, 29. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986.
F. Kasch, Modules and Rings, Academic Press Inc. (London) LTD. 1982.
Raggi, Francisco, Rincón, Hugo, Signoret, Carlos, On some classes of R-modules and congruences in R-tors. Comm. Algebra 27 (1999), no. 2, 889–901.
Raggi, Francisco, Ríos, José, Rincón, Hugo, Fernández-Alonso, Rogelio, Signoret, Carlos, The lattice structure of preradicals. Comm. Algebra 30 (2002), no. 3, 1533–1544.
Raggi, Francisco, Ríos, José, Rincón, Hugo, Fernández-Alonso, Rogelio, Signoret, Carlos, The lattice structure of preradicals II. Partitions. J. Algebra Appl. 1 (2002), no. 2, 201–214.
Raggi, Francisco, Ríos, José, Rincón, Hugo, Fernández-Alonso, Rogelio, Signoret, Carlos, The lattice structure of preradicals III. Operators. J. Pure Appl. Algebra 190 (2004), no. 1–3, 251–265.
Raggi, Francisco, Ríos, José, Rincón, Hugo, Fernández-Alonso, Rogelio, Signoret, Carlos, Prime and irreducible preradicals. J. Algebra Appl. 4 (2005), no. 4, 451–466.
Raggi, Francisco, Ríos, José, Wisbauer, Robert, The lattice structure of hereditary pretorsion classes. Comm. Algebra 29(2001), no. 1, 131–140.
Raggi, Francisco, Ríos Montes, José, Wisbauer, Robert, Coprime preradicals and modules. J. Pure Appl. Algebra 200 (2005), no. 1–2, 51–69.
Raggi, Francisco F., Signoret P., Carlos J.E., Serre subcategories of R-mod. Comm. Algebra 24 (1996), no. 9, 2877–2886.
R. Bronowitz and M. Teply, Torsion theories of simple type, J. Pure Appl. Algebra 3 (1973), 329–336.
B. Stenström, Rings of Quotients, Springer-Verlag, New York, 1975.
Y. Zhou, The Lattice of natural classes of modules, Comm. Algebra 24 (5) (1996) 1637–1648.
Y. Zhou, Decomposing modules into direct sums of submodules with types. J. Pure Appl. Algebra 138 (1999), no. 1, 83–97.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
García, A.A., Rincón Mejía, H.A., Montes, J.R. (2010). On Big Lattices of Classes of R-modules Defined by Closure Properties. In: Van Huynh, D., López-Permouth, S.R. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0286-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0286-0_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0285-3
Online ISBN: 978-3-0346-0286-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)