KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

摘要

We show that the endomorphismrings of kernels ker ϕ of non-injectivenmorphisms ϕ between indecomposable injective modules are either local or have twonmaximal ideals, the module ker ϕ is determined up to isomorphism by two invariantsncalledmonogeny class and upper part, and a weak formof theKrull–Schmidt theoremnholds for direct sums of these kernels.We prove with an example that our pathologicalndecompositions actually take place. We show that a direct sum of n kernels ofnmorphisms between injective indecomposable modules can have exactly n! pairwisennon-isomorphic direct-sum decompositions into kernels of morphisms of the samentype. If ER is an injective indecomposable module and S is its endomorphism ring,nthe duality Hom(−,ER) transforms kernels of morphisms ER → ER into cyclicallynpresented left modules over the local ring S, sending the monogeny class into thenepigeny class and the upper part into the lower part.