Kernels of Morphisms Between Indecomposable Injective Modules

Abstract

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).

Semi-invariant subrings

“…Generalizations of this theorem and counterexamples of some natural variations are widely studied (e.g. see [14], [3], [1], [12] and also [11]) and there has been considerable interest in finding rings over which all finitely presented modules have a Krull-Schmidt decomposition (e.g. [28, §6], [5], [23], [24], [30]; theorem 8.3(iii) below generalizes all these references except the last).…”

“…Both conditions are included in being semiperfect and quasi-π ∞ -regular. In addition, Vámos proved in [30,] that all finitely generated or torsionfree of finite rank modules rank over a Henselian integral domain 12 have semiperfect endomorphism ring. Results of similar flavor were also obtained in [13], where it is shown that the endomorphism ring of a f.p.…”

Direct Sums of Infinitely Many Kernels

“…The notation that will be used throughout this paper is the same notation as in our previous paper [8]. Let E 1 , E 2 , E 1 , and E 2 be indecomposable injective right modules over an arbitrary ring R, and let ϕ : E 1 → E 2 and ϕ : E 1 → E 2 be noninjective morphisms.…”

“…Let A and B be two modules. Following the terminology introduced in [5,8]: by Bumby's theorem [3]. By [8, Proposition 4.1], ker f and ker g have the same monogeny class if and only if the cyclically presented modules corresponding to ker f and ker g via an exact contravariant functor have the same epigeny class.…”

Uniqueness of Decomposition, Factorisations, G-Groups and Polynomials

“…11. (Weak Krull-Schmidt Theorem for kernels of morphisms between indecomposable injective modules,[19, Theorem 2.7] and[14]) Let ϕ i : E i,0 → E i,1 (i = 1, 2, .. . , n) and ϕ j : E j,0 → E j,1 (j = 1, 2, .…”