Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T07:00:42.106Z Has data issue: false hasContentIssue false
  • English
  • Français

GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

Published online by Cambridge University Press:  24 June 2010

TOMA ALBU
TOMA ALBU*
Affiliation:
‘Simion Stoilow’ Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-010145 Bucharest 1, Romania e-mail: Toma.Albu@imar.ro
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

Keywords

16P6016S9018E1506C0506B35
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue A: Rings and Modules in Honour of Patrick F. Smith's 65th Birthday , July 2010 , pp. 19 - 32
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Albu, T., Gabriel dimension of partially ordered sets (I), Bull. Math. Soc. Sci. Math. Roumanie 28 (76) (1984), 99–108.Google Scholar
2.Albu, T., Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (I), Bull. Math. Soc. Sci. Math. Roumanie 52 (100) (2009), 393–419.Google Scholar
3.Albu, T., A seventy years jubilee: The Hopkins-Levitzki Theorem, in Ring and module theory, trends in mathematics (Albu, T., Birkenmeier, G. F., Erdoğan, A. and Tercan, A., Editors) (Birkhäuser, Basel, 2010), 1–26.CrossRefGoogle Scholar
4.Albu, T., Iosif, M. and Teply, M. L., Modular QFD lattices with applications to Grothendieck categories and torsion theories, J. Algebra Appl. 3 (2004), 391–410.CrossRefGoogle Scholar
5.Albu, T., Iosif, M. and Teply, M. L., Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005), 52–79.CrossRefGoogle Scholar
6.Albu, T. and Năstăsescu, C., Décompositions primaires dans les catégories de Grothendieck commutatives (I), J. Reine Angew. Math. 280 (1976), 172–194.Google Scholar
7.Albu, T. and Năstăsescu, C., Relative finiteness in module theory (Marcel Dekker, Inc., New York and Basel, 1984).Google Scholar
8.Albu, T. and Rizvi, S. T., Chain conditions on quotient finite dimensional modules, Commun. Algebra 29 (2001), 1909–1928.CrossRefGoogle Scholar
9.Albu, T. and Smith, P. F., Localization of modular lattices, Krull dimension, and the Hopkins–Levitzki Theorem (I), Math. Proc. Camb. Phil. Soc. 120 (1996), 87–101.CrossRefGoogle Scholar
10.Albu, T. and Smith, P. F., Localization of modular lattices, Krull dimension, and the Hopkins–Levitzki Theorem (II), Commun. Algebra 25 (1997), 1111–1128.CrossRefGoogle Scholar
11.Albu, T. and Smith, P. F., Corrigendum and addendum to ‘Localization of Modular Lattices, Krull dimension, and the Hopkins–Levitzki Theorem (II)’, Commun. Algebra 29 (2001), 3677–3682.CrossRefGoogle Scholar
12.Albu, T. and Smith, P. F., Primal, completely irreducible, and primary meet decompositions in modules, Preprint Series of the Institute of Mathematics of the Romanian Academy, Preprint nr. 1/2009.Google Scholar
13.Albu, T. and Smith, P. F., Primality, irreducibility, and complete irreducibility in modules over commutative rings, Rev. Roumaine Math. Pures Appl. 54 (2009), 275–286.Google Scholar
14.Albu, T. and Van Den Berg, J., An indecomposable non-locally finitely generated Grothendieck category with simple objects, J. Algebra 321 (2009), 1538–1545.CrossRefGoogle Scholar
15.Birkhoff, G., Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764–768.CrossRefGoogle Scholar
16.Birkhoff, G., Lattice theory, 3rd ed. (American Mathematical Society, Providence, RI, 1967).Google Scholar
17.Camillo, V. P., Modules whose quotients have finite Goldie Dimension, Pacific J. Math. 69 (1977), 337–338.CrossRefGoogle Scholar
18.Crawley, P. and Dilworth, R. P., Algebraic theory of lattices, (Prentice-Hall, Englewood Cliffs, NJ, 1973).Google Scholar
19.Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules (Longman, Harlow, 1990).Google Scholar
20.Faith, C., Injective modules and injective quotient rings, Lecture Notes in Pure and Applied Mathematics (Marcel Dekker, Inc., New York and Basel, 1982).Google Scholar
21.Faith, C., Quotient finite dimensional modules with ACC on subdirectly irreducible submodules are Noetherian, Commun. Algebra 27 (1999), 1807–1810.CrossRefGoogle Scholar
22.Faith, C., Rings and things and a fine array of twentieth century associative algebra, 2nd ed. (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
23.Fuchs, L., Infinite Abelian groups, vol. I (Academic Press, New York and London, 1970).Google Scholar
24.Golan, J. S., Torsion theories (Pitman/Longman, New York, 1986).Google Scholar
25.Goodearl, K. and Zimmermann-Huisgen, B., Lengths of submodule chains versus Krull dimension in non-Noetherian rings, Math. Z. 191 (1986), 519–527.CrossRefGoogle Scholar
26.Grätzer, G., General lattice theory, 2nd ed. (Birkhäuser Verlag, Basel Boston Berlin, 2003).Google Scholar
27.Grzeszczuk, P. and Puczyłowski, E. R., On Goldie and dual Goldie dimension, J. Pure Appl. Algebra 31 (1984), 47–54.CrossRefGoogle Scholar
28.Herbera, D. and Shamsuddin, A., Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), 3593–3600.CrossRefGoogle Scholar
29.Huynh, D. V., Dung, N. V. and Smith, P. F., A characterization of rings with Krull dimension, J. Algebra 32 (1990), 104–112.CrossRefGoogle Scholar
30.Lau, W. G., Teply, M. L. and Boyle, A. K., The deviation, density, and depth of partially ordered sets, J. Pure Appl. Algebra 60 (1989), 253–268.CrossRefGoogle Scholar
31.Lemonnier, B., Déviation des ensembles et groupes abéliens totalement ordonnés, Bull. Sci. Math. 2e série 96 (1972), 289–303.Google Scholar
32.Lemonnier, B., Dimension de Krull et codéviation. Application au théorème d'Eakin, Commun. Algebra 16 (1978), 1647–1665.CrossRefGoogle Scholar
33.McConnell, J. C. and Robson, J. C., Noncommutative noetherian rings (John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1987).Google Scholar
34.Miller, R. W. and Teply, M. L., The descending chain condition relative to a torsion theory, Pacific J. Math. 83 (1979), 207–220.CrossRefGoogle Scholar
35.Năstăsescu, C. and Oystaeyen, F. Van, Dimensions of ring theory (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987).CrossRefGoogle Scholar
36.Puczyłowski, E. R., Linear properties of Goldie dimension of modules and modular lattices, Glasgow Math. J. 52A (2010).Google Scholar
37.Rosenstein, J. G., Linear orderings (Academic Press, New York, 1982).Google Scholar
38.Sarath, B. and Varadarajan, K., Dual Goldie dimension – II, Commun. Algebra 7 (1979), 1885–1899.CrossRefGoogle Scholar
39.Stenström, B., Rings of quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
40.Varadarajan, K., Dual Goldie dimension, Commun. Algebra 7 (1979), 565–510.CrossRefGoogle Scholar
41.Varadarajan, K., Properties of endomorphism rings, Acta Math. Hungar. 74 (1997), 83–92.CrossRefGoogle Scholar
42.Wisbauer, R., Foundations of module and ting theory (Gordon and Breach Science Publishers, Philadelphia, Reading, Paris, Tokyo, Melbourne, 1991).Google Scholar