Geodesic
KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES
Glasgow mathematical journal, Tome 52 (2010) no. A, pp. 69-82

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DOI

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.
DOI : 10.1017/S0017089510000170
Mots-clés : 15A33 
FACCHINI, ALBERTO; ECEVIT, ŞULE; KOŞAN, M. TAMER. KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES. Glasgow mathematical journal, Tome 52 (2010) no. A, pp. 69-82. doi: 10.1017/S0017089510000170
@article{10_1017_S0017089510000170,
     author = {FACCHINI, ALBERTO and ECEVIT, \c{S}ULE and KO\c{S}AN, M. TAMER},
     title = {KERNELS {OF} {MORPHISMS} {BETWEEN} {INDECOMPOSABLE} {INJECTIVE} {MODULES}},
     journal = {Glasgow mathematical journal},
     pages = {69--82},
     year = {2010},
     volume = {52},
     number = {A},
     doi = {10.1017/S0017089510000170},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000170/}
}
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