Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-23T15:02:14.548Z Has data issue: false hasContentIssue false
  • English
  • Français

DIRECT SUMS OF INFINITELY MANY KERNELS

Part of: Local rings and generalizations Modules, bimodules and ideals

Published online by Cambridge University Press:  23 November 2010

ŞULE ECEVIT ,
ALBERTO FACCHINI  and
M. TAMER KOŞAN
ŞULE ECEVIT
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey (email: secevit@gyte.edu.tr)
ALBERTO FACCHINI*
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, 35121 Padova, Italy (email: facchini@math.unipd.it)
M. TAMER KOŞAN
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey (email: mtkosan@gyte.edu.tr)
*
✉For correspondence; e-mail: facchini@math.unipd.it
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.

Let 𝒦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1→E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

Keywords

moduledirect sum decompositionsemilocal endomorphism ring

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 16L30: Noncommutative local and semilocal rings, perfect rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 2 , October 2010 , pp. 199 - 214
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

Alberto Facchini was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (Prin 2007 ‘Rings, algebras, modules and categories’) and by the Università di Padova (Progetto di Ricerca di Ateneo CPDA071244/07).

References

[1]Amini, B., Amini, A. and Facchini, A., ‘Equivalence of diagonal matrices over local rings’, J. Algebra 320 (2008), 1288–1310.CrossRefGoogle Scholar
[2]Amini, B., Amini, A. and Facchini, A., ‘Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals’, manuscript (2010).CrossRefGoogle Scholar
[3]Bumby, R. T., ‘Modules which are isomorphic to submodules of each other’, Arch. Math. 16 (1965), 184–185.CrossRefGoogle Scholar
[4]Dung, N. V. and Facchini, A., ‘Weak Krull–Schmidt for infinite direct sums of uniserial modules’, J. Algebra 193 (1997), 102–121.CrossRefGoogle Scholar
[5]Facchini, A., ‘Krull–Schmidt fails for serial modules’, Trans. Amer. Math. Soc. 348 (1996), 4561–4575.Google Scholar
[6]Facchini, A., Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics, 167 (Birkhäuser, Basel, 1998).Google Scholar
[7]Facchini, A., ‘Injective modules, spectral categories, and applications’, in: Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications, Contemporary Mathematics, 456 (eds. Jain, S. K. and Parvathi, S.) (American Mathematical Society, Providence, RI, 2008), pp. 1–17.Google Scholar
[8]Facchini, A., Ecevit, Ş. and Tamer Koşan, M., ‘Kernels of morphisms between indecomposable injective modules’, Glasgow Math. J. (2010), to appear.CrossRefGoogle Scholar
[9]Facchini, A. and Girardi, N., ‘Couniformly presented modules and dualities’, in: Advances in Ring Theory, Trends in Mathematics (eds. Huynh, Dinh Van and López Permouth, Sergio R.) (Birkhäuser, Basel, 2010), pp. 149–163.CrossRefGoogle Scholar
[10]Facchini, A. and Příhoda, P., ‘Representations of the category of serial modules of finite Goldie dimension’, in: Models, Modules and Abelian Groups (eds. Göbel, R. and Goldsmith, B.) (de Gruyter, Berlin, 2008), pp. 463–486.Google Scholar
[11]Facchini, A. and Příhoda, P., ‘Factor categories and infinite direct sums’, Int. Electron. J. Algebra 5 (2009), 1–34.Google Scholar
[12]Gabriel, P. and Oberst, U., ‘Spektralkategorien und reguläre Ringe im Von-Neumannschen Sinn’, Math. Z. 82 (1966), 389–395.Google Scholar
[13]Herzog, I., ‘Contravariant functors on the category of finitely presented modules’, Israel J. Math. 167 (2008), 347–410.Google Scholar
[14]Příhoda, P., ‘A version of the weak Krull–Schmidt theorem for infinite direct sums of uniserial modules’, Comm. Algebra 34 (2006), 1479–1487.CrossRefGoogle Scholar
[15]Puninski, G., ‘Some model theory over a nearly simple uniserial domain and decompositions of serial modules’, J. Pure Appl. Algebra 163 (2001), 319–337.CrossRefGoogle Scholar