GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

ALBU, TOMA

Glasgow mathematical journal, Tome 52 (2010) no. A, pp. 19-32

Résumé

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.

Détail

DOI : 10.1017/S0017089510000285

Mots-clés : 16P60, 16S90, 18E15, 06C05, 06B35

Bibliographie

[1] 1.Albu, T., Gabriel dimension of partially ordered sets (I), Bull. Math. Soc. Sci. Math. Roumanie 28 (76) (1984), 99–108. Google Scholar

[2] 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] 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. Google Scholar | DOI

[4] 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. Google Scholar | DOI

[5] 5.Albu, T., Iosif, M. and Teply, M. L., Dual Krull dimension and quotient finite dimensionality, J. Algebra 284 (2005), 52–79. Google Scholar | DOI

[6] 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] 7.Albu, T. and Năstăsescu, C., Relative finiteness in module theory (Marcel Dekker, Inc., New York and Basel, 1984). Google Scholar

[8] 8.Albu, T. and Rizvi, S. T., Chain conditions on quotient finite dimensional modules, Commun. Algebra 29 (2001), 1909–1928. Google Scholar | DOI

[9] 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. Google Scholar | DOI

[10] 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. Google Scholar | DOI

[11] 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. Google Scholar | DOI

[12] 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] 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] 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. Google Scholar | DOI

[15] 15.Birkhoff, G., Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764–768. Google Scholar | DOI

[16] 16.Birkhoff, G., Lattice theory, 3rd ed. (American Mathematical Society, Providence, RI, 1967). Google Scholar

[17] 17.Camillo, V. P., Modules whose quotients have finite Goldie Dimension, Pacific J. Math. 69 (1977), 337–338. Google Scholar | DOI

[18] 18.Crawley, P. and Dilworth, R. P., Algebraic theory of lattices, (Prentice-Hall, Englewood Cliffs, NJ, 1973). Google Scholar

[19] 19.Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules (Longman, Harlow, 1990). Google Scholar

[20] 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] 21.Faith, C., Quotient finite dimensional modules with ACC on subdirectly irreducible submodules are Noetherian, Commun. Algebra 27 (1999), 1807–1810. Google Scholar | DOI

[22] 22.Faith, C., Rings and things and a fine array of twentieth century associative algebra, 2nd ed. (American Mathematical Society, Providence, RI, 2004). Google Scholar | DOI

[23] 23.Fuchs, L., Infinite Abelian groups, vol. I (Academic Press, New York and London, 1970). Google Scholar

[24] 24.Golan, J. S., Torsion theories (Pitman/Longman, New York, 1986). Google Scholar

[25] 25.Goodearl, K. and Zimmermann-Huisgen, B., Lengths of submodule chains versus Krull dimension in non-Noetherian rings, Math. Z. 191 (1986), 519–527. Google Scholar | DOI

[26] 26.Grätzer, G., General lattice theory, 2nd ed. (Birkhäuser Verlag, Basel Boston Berlin, 2003). Google Scholar

[27] 27.Grzeszczuk, P. and Puczyłowski, E. R., On Goldie and dual Goldie dimension, J. Pure Appl. Algebra 31 (1984), 47–54. Google Scholar | DOI

[28] 28.Herbera, D. and Shamsuddin, A., Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), 3593–3600. Google Scholar | DOI

[29] 29.Huynh, D. V., Dung, N. V. and Smith, P. F., A characterization of rings with Krull dimension, J. Algebra 32 (1990), 104–112. Google Scholar | DOI

[30] 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. Google Scholar | DOI

[31] 31.Lemonnier, B., Déviation des ensembles et groupes abéliens totalement ordonnés, Bull. Sci. Math. 2e série 96 (1972), 289–303. Google Scholar

[32] 32.Lemonnier, B., Dimension de Krull et codéviation. Application au théorème d'Eakin, Commun. Algebra 16 (1978), 1647–1665. Google Scholar | DOI

[33] 33.McConnell, J. C. and Robson, J. C., Noncommutative noetherian rings (John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1987). Google Scholar

[34] 34.Miller, R. W. and Teply, M. L., The descending chain condition relative to a torsion theory, Pacific J. Math. 83 (1979), 207–220. Google Scholar | DOI

[35] 35.Năstăsescu, C. and Oystaeyen, F. Van, Dimensions of ring theory (D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, Tokyo, 1987). Google Scholar | DOI

[36] 36.Puczyłowski, E. R., Linear properties of Goldie dimension of modules and modular lattices, Glasgow Math. J. 52A (2010). Google Scholar

[37] 37.Rosenstein, J. G., Linear orderings (Academic Press, New York, 1982). Google Scholar

[38] 38.Sarath, B. and Varadarajan, K., Dual Goldie dimension – II, Commun. Algebra 7 (1979), 1885–1899. Google Scholar | DOI

[39] 39.Stenström, B., Rings of quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975). Google Scholar | DOI

[40] 40.Varadarajan, K., Dual Goldie dimension, Commun. Algebra 7 (1979), 565–510. Google Scholar | DOI

[41] 41.Varadarajan, K., Properties of endomorphism rings, Acta Math. Hungar. 74 (1997), 83–92. Google Scholar | DOI

[42] 42.Wisbauer, R., Foundations of module and ting theory (Gordon and Breach Science Publishers, Philadelphia, Reading, Paris, Tokyo, Melbourne, 1991). Google Scholar