Injective Modules, Induction Maps and Endomorphism Rings

AbstractThe study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

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Endomorphism rings of completely pure-injective modules

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-96-03240-6 ◽

1996 ◽

Vol 124 (8) ◽

pp. 2301-2309 ◽

Cited By ~ 7

Author(s):

José L. Gómez Pardo ◽

Pedro A. Guil Asensio

Keyword(s):

Endomorphism Rings ◽

Injective Modules

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REGULAR ENDOMORPHISM RINGS OF INJECTIVE MODULES

Honam Mathematical Journal ◽

10.5831/hmj.2009.31.4.529 ◽

2009 ◽

Vol 31 (4) ◽

pp. 529-540

Author(s):

Sun-Ah Kim ◽

Soon-Sook Bae

Keyword(s):

Endomorphism Rings ◽

Injective Modules

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Endomorphism Rings of Quasi-Injective Modules

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-086-3 ◽

1968 ◽

Vol 20 ◽

pp. 895-903 ◽

Cited By ~ 16

Author(s):

B. L. Osofsky

Keyword(s):

Jacobson Radical ◽

Essential Extension ◽

Endomorphism Rings ◽

Injective Modules

Y. Utumi (14 and 15) obtained some interesting results on self-injective rings. He showed that, if R is right self-injective, then so is R/J, where J is the Jacobson radical of R. Also, if R is right self-injective and regular, then R is left self-injective for any set of orthogonal idempotents is an essential extension of . This note extends these results to endomorphism rings of quasi-injective modules.

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SP-INJECTIVITY OF MODULES AND RINGS

Asian-European Journal of Mathematics ◽

10.1142/s1793557112500532 ◽

2012 ◽

Vol 05 (04) ◽

pp. 1250053

Author(s):

A. J. Gupta ◽

B. M. Pandeya ◽

A. K. Chaturvedi

Keyword(s):

Injective Module ◽

Endomorphism Rings ◽

Injective Modules

Let >M and N be two R-modules. NR is called singular M-p-injective if for every singular M-cyclic submodule X of MR, every homomorphism from X to N can be extended to a homomorphism from M to N. MR is quasi-singular prinicipally injective if M is a singular M-p-injective module. It is shown that a ring R is right non-singular if and only if every right R-module is singular R-p-injective if and only if factors of singular R-p-injective modules are singular R-p-injective. A singular R-module M is injective if and only if M is N-sp-injective for every R-module N. Finally, we characterize quasi-sp-injective modules in terms of their endomorphism rings.

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ENDOMORPHISM RINGS OF QUASI-PRINCIPALLY INJECTIVE MODULES

Communications in Algebra ◽

10.1081/agb-100002110 ◽

2001 ◽

Vol 29 (4) ◽

pp. 1437-1443 ◽

Cited By ~ 5

Author(s):

Nguyen Van Sanh ◽

Kar Ping Shum

Keyword(s):

Endomorphism Rings ◽

Injective Modules

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Structure of Some Noetherian Injective Modules

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-097-2 ◽

1980 ◽

Vol 32 (6) ◽

pp. 1277-1287 ◽

Cited By ~ 1

Author(s):

B. Sarath

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Finitely Generated ◽

Endomorphism Rings ◽

Commutative Case ◽

Noetherian Module ◽

Direct Sum ◽

Injective Modules ◽

Semiprime Rings ◽

Noetherian Modules

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. IfQis an injective indecomposable noetherian module, thenQcontains a non-zero submoduleQ0such that the endomorphism rings ofQ0and all its submodules are skewfields. Over a commutative ring, such aQ0is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.

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On Hermitian endomorphism rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498817502206 ◽

2017 ◽

Vol 16 (11) ◽

pp. 1750220

Author(s):

Wanru Zhang

Keyword(s):

Endomorphism Ring ◽

Necessary Conditions ◽

Exchange Property ◽

Projective Modules ◽

Endomorphism Rings ◽

Dual Problems ◽

Injective Modules ◽

Sufficient And Necessary Conditions

Let [Formula: see text] be a right [Formula: see text]-module with finite exchange property and let [Formula: see text] be its endomorphism ring. In this paper, some sufficient and necessary conditions for [Formula: see text] to be a Hermitian ring are given. Moreover, we investigate Hermitian endomorphism rings of quasi-projective modules by means of completions of diagrams. The dual problems for quasi-injective modules are also studied.

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KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

Glasgow Mathematical Journal ◽

10.1017/s0017089510000170 ◽

2010 ◽

Vol 52 (A) ◽

pp. 69-82 ◽

Cited By ~ 13

Author(s):

ALBERTO FACCHINI ◽

ŞULE ECEVIT ◽

M. TAMER KOŞAN

Keyword(s):

Local Ring ◽

Endomorphism Ring ◽

Weak Form ◽

Indecomposable Module ◽

Endomorphism Rings ◽

Direct Sums ◽

Indecomposable Modules ◽

Direct Sum ◽

Injective Modules ◽

Maximal Ideals

AbstractWe 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 ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

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