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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: [email protected]
ŞULE ECEVIT
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey e-mail: [email protected], [email protected]
M. TAMER KOŞAN
Affiliation:
Department of Mathematics, Gebze Institute of Technology, Çayirova Campus, 41400 Gebze-Kocaeli, Turkey e-mail: [email protected], [email protected]
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Abstract

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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

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