Kernels of morphisms between indecomposable injective modules

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. © 2010, Glasgow Mathematical Journal Trust. All rights reserv...