Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:20:56.346Z Has data issue: false hasContentIssue false
  • English
  • Français

KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

Published online by Cambridge University Press:  24 June 2010

ALBERTO FACCHINI ,
ŞULE ECEVIT  and
M. TAMER KOŞAN
ALBERTO FACCHINI
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, 35131 Padova, Italy e-mail: facchini@math.unipd.it
ŞULE ECEVIT
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey e-mail: secevit@gyte.edu.tr, mtkosan@gyte.edu.tr
M. TAMER KOŞAN
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey e-mail: secevit@gyte.edu.tr, mtkosan@gyte.edu.tr
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.

We show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum of n kernels of morphisms between injective indecomposable modules can have exactly n! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. If ER is an injective indecomposable module and S is its endomorphism ring, the duality Hom(−, ER) transforms kernels of morphisms ER → ER into cyclically presented left modules over the local ring S, sending the monogeny class into the epigeny class and the upper part into the lower part.

Keywords

15A33
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. 69 - 82
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Anderson, F. W. and Fuller, K. R., Rings and categories of modules, 2nd edn., GTM, vol. 13 (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
2.Amini, B., Amini, A. and Facchini, A., Equivalence of diagonal matrices over local rings, J. Algebra 320 (2008), 1288–1310.CrossRefGoogle Scholar
3.Bumby, R. T., Modules which are isomorphic to submodules of each other, Arch. Math. 16 (1965), 184–185.CrossRefGoogle Scholar
4.Facchini, A., Krull–Schmidt fails for serial modules, Trans. Amer. Math. Soc. 348 (1996), 4561–4575.CrossRefGoogle Scholar
5.Facchini, A., Module theory. Endomorphism rings and direct sum decompositions in some classes of modules, Progress in Mathematics, vol. 167 (Birkhäuser Verlag, Basel, Switzerland, 1998).Google Scholar
6.Facchini, A. and Herbera, D., Local morphisms and modules with a semilocal endomorphism ring, Algebr. Represent. Theory 9 (2006), 403–422.CrossRefGoogle Scholar
7.Gill, D. T., Almost maximal valuation rings, J. Lond. Math. Soc. 4 (2) (1971), 140–146.CrossRefGoogle Scholar
8.Matlis, E., Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528.CrossRefGoogle Scholar
9.Shores, T. S. and Lewis, J. W., Serial modules and endomorphism rings, Duke Math. J. 41 (1974), 889–909.CrossRefGoogle Scholar