Modules which are isomorphic to submodules of each other

Bumby, Richard T.

doi:10.1007/bf01220018

Cited by 53 publications

(35 citation statements)

References 2 publications

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“…While existence of minimal cotilting modules is a non-trivial fact, their uniqueness up to isomorphism follows easily from their pure-injectivity [13] and from a classic result of Bumby [6]; it does not require the noetherian or commutative assumption on R: Lemma 2.3. Let R be an arbitrary ring.…”

Section: Preliminariesmentioning

confidence: 99%

Cotilting modules over commutative Noetherian rings

Šťovíček

Trlifaj

Herbera

2014

Journal of Pure and Applied Algebra

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Abstract. Recently, tilting and cotilting classes over commutative noetherian rings have been classified in [2]. We proceed and, for each n-cotilting class C, construct an n-cotilting module inducing C by an iteration of injective precovers. A further refinement of the construction yields the unique minimal n-cotilting module inducing C. Finally, we consider localization: a cotilting module is called ample, if all of its localizations are cotilting. We prove that for each 1-cotilting class, there exists an ample cotilting module inducing it, but give an example of a 2-cotilting class which fails this property.

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Section: Preliminariesmentioning

confidence: 99%

Cotilting modules over commutative Noetherian rings

Šťovíček

Trlifaj

Herbera

2014

Journal of Pure and Applied Algebra

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“…Bumby's theorem [3]. Moreover, we shall see in Proposition 4.1 that 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, and ker f and ker g have the same upper part if and only if the modules corresponding to ker f and ker g via the same contravariant functor have the same lower part in the sense of [2].…”

Section: So That E(e(a)/a) ∼ = E(e(b)/b) Bymentioning

confidence: 99%

Kernels of Morphisms Between Indecomposable Injective Modules

Facchini¹,

Ecevit²,

Koşan³

2010

Glasgow Math. J.

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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).

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“…Since M and N are isomorphic to a direct summand of the other, the same happens for F E (M) and F E (N). By [3],…”

Section: (A) If E Is An Idempotent Endomorphism Of E With Image M Andmentioning

confidence: 99%