New results on C₁₁ and C₁₂ lattices with applications to Grothendieck categories and torsion theories

Abstract

In this paper, which is a cont inuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C_i) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C₁₁) or (C₁₂). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

Keywords

Modular lattice / upper continuous lattice / essential element / complement element / closed element / uniform lattice / condition (C_i) / C₁₁ lattice / C₁₂ lattice / Goldie dimension / socle / Grothendieck category / torsion theory

References

[1] Albu T. The Osofsky-Smith Theorem for modular lattices, and applications (II). Comm Algebra, 2014, 42: 2663–2683

[2] Albu T. Topics in Lattice Theory with Applications to Rings, Modules, and Categories. Lecture Notes, XXIII Brazilian Algebra Meeting, Maringá Paraná Brasil, 2014 (80 pages)

[3] Albu T. Chain Conditions in Modular Lattices with Applications to Grothendieck Categories and Torsion Theories. Monograph Series of Parana’s Mathematical Society No 1, Sociedade Paranaense de Matemática, Maringá Paraná Brasil, 2015 (134 pages)

[4] Albu T, Iosif M. The category of linear modular lattices. Bull Math Soc Sci Math Roumanie, 2013, 56(104): 33–46

[5] Albu T, Iosif M. Lattice preradicals with applications to Grothendieck categories and torsion theories. J Algebra, 2015, 444: 339–366

[6] Albu T, Iosif M, Teply M L. Modular QFD lattices with applications to Grothendieck categories and torsion theories. J Algebra Appl, 2004, 3: 391–410

[7] Albu T, Iosif M, Tercan A. The conditions (Ci) in modular lattices, and applications. J Algebra Appl, 2016, 15: (19 pages), http:dx.doi.org/10.1142/S0219498816500018

[8] Albu T, Nǎstǎsescu C. Relative Finiteness in Module Theory. New York and Basel: Marcel Dekker, Inc, 1984

[9] Crawley P, Dilworth R P. Algebraic Theory of Lattices. Englewood Cliffs: Prentice-Hall, 1973

[10] Galvão M L, Smith P F. Chain conditions in modular lattices. Colloq Math, 1998, 76: 85–98

[11] Grzeszczuk P, Puczi_lowski E R. On finiteness conditions of modular lattices. Comm Algebra, 1998, 26: 2949–2957

[12] Mohamed S H, Müller B J. Continuous and Discrete Modules. Cambridge: Cambridge University Press, 1990

[13] Nǎstǎsescu C, Van Oystaeyen F. Dimensions of Ring Theory. Dordrecht-Boston-Lancaster-Tokyo: D Reidel Publishing Company, 1987

[14] Smith P F, Tercan A. Generalizations of CS-modules. Comm Algebra, 1993, 21: 1809–1847

[15] Stenström B. Rings of Quotients. Berlin-Heidelberg-New York: Springer-Verlag, 1975

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Higher Education Press and Springer-Verlag Berlin Heidelberg