Local Morphisms and Modules with a Semilocal Endomorphism Ring

Facchini, Alberto; Herbera, Dolors

doi:10.1007/s10468-006-9011-8

Cited by 29 publications

(24 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main sources of examples are the papers [59] and [38]. The following modules have a semilocal endomorphism rings:…”

Section: Modules With a Semilocal Endomorphism Ringmentioning

confidence: 99%

Direct-sum decompositions of modules with semilocal endomorphism rings

Facchini

2012

Bull. Math. Sci.

Self Cite

View full text Add to dashboard Cite

According to the classical Krull-Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J (E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of Communicated by S.K. Jain.Partially supported by Università di Padova (Progetto di ricerca di Ateneo CPDA105885/10 "Differential graded categories" and Progetto ex 60% "Anelli e categorie di moduli").

show abstract

“…The main sources of examples are the papers [59] and [38]. The following modules have a semilocal endomorphism rings:…”

Section: Modules With a Semilocal Endomorphism Ringmentioning

confidence: 99%

Direct-sum decompositions of modules with semilocal endomorphism rings

Facchini

2012

Bull. Math. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This contradiction proves the claim. (Note that the proof of this claim is modelled on the proof of [6]. The most important sources of ideas for this paper have been [4] and [6].)…”

Section: The Endomorphism Ringmentioning

confidence: 99%

Kernels of Morphisms Between Indecomposable Injective Modules

Facchini¹,

Ecevit²,

Koşan³

2010

Glasgow Math. J.

Self Cite

View full text Add to dashboard Cite

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).

show abstract

“…They studied local morphisms in the setting of general noncommutative rings. For instance, they proved that if A is an object in a Grothendieck category C, such that A has finite Goldie dimension and that every monomorphism A → A is an isomorphism, then its endomorphism ring End C (A) is semilocal [7,Corollary 4.5]. This result is a slight generalization of [9,Theorem 3(1)].…”

Section: Introductionmentioning

confidence: 99%

“…The motivation for this paper comes from the work by Facchini and Herbera [7] on semilocal endomorphism ring. They studied local morphisms in the setting of general noncommutative rings.…”

Section: Introductionmentioning

confidence: 99%

“…For the proof of this result, they use the theorem by Camps and Dicks [4, Theorem 1] which characterizes semilocal rings in terms of local morphisms. Then they show that local morphisms arise naturally in the construction of the spectral category Spec − C of an arbitrary Grothendieck category C. The present paper contains an extension of [7,Corollary 4.5] In Section 2, we introduce a Goldie dimension (rank) concept for finitely accessible additive categories. In Section 3, we give our main result, namely, we prove that if A is an object in a finitely accessible additive category A such that A has finite pure Goldie dimension and that every pure monomorphism A → A is an isomorphism, then its endomorphism ring End A (A) is semilocal.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation