Abstract. 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 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.2000 Mathematics Subject Classification. 15A33. Introduction.In 1996, the first author described the behaviour, as far as direct sums are concerned, of modules M R of Goldie dimension one and dual Goldie dimension one [4]. The endomorphism rings of these modules M R are either local or have two maximal ideals, the module M R is determined up to isomorphism by two invariants called monogeny class and epigeny class, and a weak form of the KrullSchmidt theorem holds for direct sums of these modules. In 2008 it was discovered [2] that a second class of modules has exactly the same behaviour. It is the class of cyclically presented modules over a local ring. The endomorphism ring of a cyclically presented module N R over a local ring R is either local or has two maximal ideals, the module N R is determined up to isomorphism by its epigeny class and another invariant, called lower part, and a weak form of the Krull-Schmidt theorem holds for direct sums of these modules as well.In this paper, we present a third class of modules with the same behaviour. They are the kernels of morphisms ϕ between indecomposable injective modules. The endomorphism ring of such a kernel ker ϕ is either local or has two maximal ideals, the module ker ϕ is determined up to isomorphism by its monogeny class and a second invariant, called upper part, and a weak form of the Krull-Schmidt theorem similar to that of the previous two classes also holds for direct sums of these kernels (Theorem 2.7).
A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M . We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.
Let K be the class of all right R-modules that are kernels of nonzero homomorphisms ϕ : E 1 → E 2 for some pair of indecomposable injective right R-modules E 1 , E 2 . In a previous paper, we completely characterized when two direct sums A 1 ⊕ · · · ⊕ A n and B 1 ⊕ · · · ⊕ B m of finitely many modules A i and B j in K are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many A i and B j in K. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class K with the class U of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).2000 Mathematics subject classification: primary 16D70; secondary 16E05, 16L30.
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