KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

We show that the endomorphism rings of kernels ker phi of non-injective morphisms phi between indecomposable injective modules are either local or have two maximal ideals, the module ker phi 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 E(R) is an injective indecomposable module and S is its endomorphism ring, the duality Hom(-, E(R)) transforms kernels of morphisms E(R) -> E(R) 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.