On τ-Extending Modules
01 Feb 2012-Mediterranean Journal of Mathematics (SP Birkhäuser Verlag Basel)-Vol. 9, Iss: 1, pp 129-142
TL;DR: In this article, a generalization of extending (CS) modules by using the concept of τ-large submodule was introduced, and the authors give some properties of this class of modules and study their relationship with τ-closed, τ-complement submodules and the other generalisation of extending modules (τ-complemented and τ-CS, s−τ-CS modules).
Abstract: Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ-large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ-closed, τ-complement submodules and the other generalization of extending modules (τ-complemented, τ-CS, s−τ-CS modules). We are also interested in determining when a τ-divisible module is τ-extending. For a τ-extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that \(M = \tau (M)\,\oplus\,K\) and K is τ (M)-injective. We also treat when a direct sum of τ-extending modules is τ-extending.
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TL;DR: In this paper, it was shown that a prime right nonsingular ring (in particular, a simple ring) is right self-injective if it is invariant under automorphisms of its injective hull.
Abstract: It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated.
54 citations
Cites background from "On τ-Extending Modules"
...A detailed treatment of the above concepts and other related facts can be found in [6] and [10]....
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TL;DR: In this article, it was shown that for a nonsingular extending module M, E (M ) ⊕ M is always a Rickart module, if and only if M i is M j -injective for all i j ∈ I = { 1, 2, …, n }.
45 citations
Cites background from "On τ-Extending Modules"
...Note that for any n ∈ N, Z(n) is an extending and Baer Z-module, Z(N) is a Baer but not an extending Z-module (Page 56, [9]), and Z(R) is a Rickart but neither a Baer nor an extending Z-module (Remark 2....
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TL;DR: In this paper, the authors studied the lattice theoretic properties of the profiles of a ring R and showed that the right p-profile is isomorphic to an interval of linear filters of right ideals of R, and is therefore modular and coatomic.
30 citations
TL;DR: In this article, a module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. The notion of extending is introduced to describe the behavior of modules with respect to direct sums and direct summands.
Abstract: A module M is called an extending (or CS) module provided that every submodule of M is essential in a direct summand of M. We call a module 𝒞-extending if every member of the set 𝒞 is essential in a direct summand where 𝒞 is a subset of the set of all submodules of M. Our focus is the behavior of the 𝒞-extending modules with respect to direct sums and direct summands. By obtaining various well-known results on extending modules and generalizations as corollaries of our results, we show that the 𝒞-extending concept provides a unifying framework for many generalizations of the extending notion. Moreover, by applying our results to various sets 𝒞, including the projection invariant submodules, the projective submodules, and torsion or torsion-free submodules of a module, we obtain new results including a characterization of the projection invariant extending Abelian groups.
25 citations
TL;DR: In this article, it was shown that a Kothe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring and determine when R is retractable.
Abstract: We carry out a study of rings R for which HomR (M;N) 6= 0 for all nonzero N ≤ MR. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Kothe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.
17 citations
Cites background from "On τ-Extending Modules"
...Any unexplained terminology, and all the basic results on rings and modules that are used in the sequel can be found in [2], [5] and [14]....
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References
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Book•
01 Jan 1999
TL;DR: In this article, Baer's Test for Injectivity is used to evaluate the suitability of a set of Injective Modules over a group of Invertible Fractional Ideals.
Abstract: 1 Free Modules, Projective, and Injective Modules.- 1. Free Modules.- 1A. Invariant Basis Number (IBN).- 1B. Stable Finiteness.- 1C. The Rank Condition.- 1D. The Strong Rank Condition.- 1E. Synopsis.- Exercises for 1.- 2. Projective Modules.- 2A. Basic Definitions and Examples.- 2B. Dual Basis Lemma and Invertible Modules.- 2C. Invertible Fractional Ideals.- 2D. The Picard Group of a Commutative Ring.- 2E. Hereditary and Semihereditary Rings.- 2F. Chase Small Examples.- 2G. Hereditary Artinian Rings.- 2H. Trace Ideals.- Exercises for 2.- 3. Injective Modules.- 3A. Baer's Test for Injectivity.- 3B. Self-Injective Rings.- 3C. Injectivity versus Divisibility.- 3D. Essential Extensions and Injective Hulls.- 3E. Injectives over Right Noetherian Rings.- 3F. Indecomposable Injectives and Uniform Modules.- 3G. Injectives over Some Artinian Rings.- 3H. Simple Injectives.- 31. Matlis' Theory.- 3J. Some Computations of Injective Hulls.- 3K. Applications to Chain Conditions.- Exercises for 3.- 2 Flat Modules and Homological Dimensions.- 4. Flat and Faithfully Flat Modules.- 4A. Basic Properties and Flatness Tests.- 4B. Flatness, Torsion-Freeness, and von Neumann Regularity.- 4C. More Flatness Tests.- 4D. Finitely Presented (f.p.) Modules.- 4E. Finitely Generated Flat Modules.- 4F. Direct Products of Flat Modules.- 4G. Coherent Modules and Coherent Rings.- 4H. Semihereditary Rings Revisited.- 41. Faithfully Flat Modules.- 4J. Pure Exact Sequences.- Exercises for 4.- 5. Homological Dimensions.- 5A. Schanuel's Lemma and Projective Dimensions.- 5B. Change of Rings.- 5C. Injective Dimensions.- 5D. Weak Dimensions of Rings.- 5E. Global Dimensions of Semiprimary Rings.- 5F. Global Dimensions of Local Rings.- 5G. Global Dimensions of Commutative Noetherian Rings.- Exercises for 5.- 3 More Theory of Modules.- 6. Uniform Dimensions, Complements, and CS Modules.- 6A. Basic Definitions and Properties.- 6B. Complements and Closed Submodules.- 6C. Exact Sequences and Essential Closures.- 6D. CS Modules: Two Applications.- 6E. Finiteness Conditions on Rings.- 6F. Change of Rings.- 6G. Quasi-Injective Modules.- Exercises for 6.- 7. Singular Submodules and Nonsingular Rings.- 7A. Basic Definitions and Examples.- 7B. Nilpotency of the Right Singular Ideal.- 7C. Goldie Closures and the Reduced Rank.- 7D. Baer Rings and Rickart Rings.- 7E. Applications to Hereditary and Semihereditary Rings.- Exercises for 7.- 8. Dense Submodules and Rational Hulls.- 8A. Basic Definitions and Examples.- 8B. Rational Hull of a Module.- 8C. Right Kasch Rings.- Exercises for 8.- 4 Rings of Quotients.- 9. Noncommutative Localization.- 9A. "The Good'.- 9B. "The Bad'.- 9C. "The Ugly".- 9D. An Embedding Theorem of A. Robinson.- Exercises for 9.- 10. Classical Rings of Quotients.- 10A. Ore Localizations.- 10B. Right Ore Rings and Domains.- 10C. Polynomial Rings and Power Series Rings.- 10D. Extensions and Contractions.- Exercises for 10.- 11. Right Goldie Rings and Goldie's Theorems.- 11A. Examples of Right Orders.- 11B. Right Orders in Semisimple Rings.- 11C. Some Applications of Goldie's Theorems.- 11D. Semiprime Rings.- 11E. Nil Multiplicatively Closed Sets.- Exercises for 11.- 12. Artinian Rings of Quotients.- 12A. Goldie's ?-Rank.- 12B. Right Orders in Right Artinian Rings.- 12C. The Commutative Case.- 12D. Noetherian Rings Need Not Be Ore.- Exercises for 12.- 5 More Rings of Quotients.- 13. Maximal Rings of Quotients.- 13A. Endomorphism Ring of a Quasi-Injective Module.- 13B. Construction of Qrmax(R).- 13C. Another Description of Qrmax(R).- 13D. Theorems of Johnson and Gabriel.- Exercises for 13.- 14. Martindale Rings of Quotients.- 14A. Semiprime Rings Revisited.- 14B. The Rings Qr(R) and Qs(R).- 14C. The Extended Centroid.- 14D. Characterizations of and Qr(R) and Qs(R).- 14E. X-Inner Automorphisms.- 14F. A Matrix Ring Example.- Exercises for 14.- 6 Frobenius and Quasi-Frobenius Rings.- 15. Quasi-Frobenius Rings.- 15A. Basic Definitions of QF Rings.- 15B. Projectives and Injectives.- 15C. Duality Properties.- 15D. Commutative QF Rings, and Examples.- Exercises for 15.- 16. Frobenius Rings and Symmetric Algebras.- 16A. The Nakayama Permutation.- 16B. Definition of a Frobenius Ring.- 16C. Frobenius Algebras and QF Algebras.- 16D. Dimension Characterizations of Frobenius Algebras.- 16E. The Nakayama Automorphism.- 16F. Symmetric Algebras.- 16G. Why Frobenius?.- Exercises for 16.- 7 Matrix Rings, Categories of Modules, and Morita Theory.- 17. Matrix Rings.- 17A. Characterizations and Examples.- 17B. First Instance of Module Category Equivalences.- 17C. Uniqueness of the Coefficient Ring.- Exercises for 17.- 18. Morita Theory of Category Equivalences.- 18A. Categorical Properties.- 18B. Generators and Progenerators.- 18C. The Morita Context.- 18D. Morita I, II, III.- 18E. Consequences of the Morita Theorems.- 18F. The Category ? [M].- Exercises for 18.- 19. Morita Duality Theory.- 19A. Finite Cogeneration and Cogenerators.- 19B. Cogenerator Rings.- 19C. Classical Examples of Dualities.- 19D. Morita Dualities: Morita I.- 19E. Consequences of Morita I.- 19F. Linear Compactness and Reflexivity.- 19G. Morita Dualities: Morita II.- Exercises for 19.- References.- Name Index.
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Book•
01 Jan 1989TL;DR: The 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra as mentioned in this paper, and it can be used as a second-year graduate text, or as a self-contained reference.
Abstract: This 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. Material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
985 citations
TL;DR: In this paper, the authors investigate two generalizations of CS-modules, i.e., if every submodule is essential in a direct sum-mand of a module, and if the module M is a CS-module or satifies (C1).
Abstract: Let R be a ring and M a right R-module. The module M is a CS-module or satifies (C1) if every submodule is essential in a direct summand of M. In this note we investigate two generalizations of CS-modules.
65 citations
"On τ-Extending Modules" refers background in this paper
...Rings for which a certain class of modules satisfies some “generalized extending property” have been studied by many authors (see [5], [15], [ 16 ])....
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TL;DR: In this article, Spectral gabriel topologies and relative singular functors are studied in the context of algebraic topologies, and the authors propose a topology based on Spectral Gabbriel topology.
Abstract: (1985). Spectral gabriel topologies and relative singular functors. Communications in Algebra: Vol. 13, No. 1, pp. 21-57.
23 citations
"On τ-Extending Modules" refers methods in this paper
...In the second part of this note, we study the properties of τ -closed submodules defined by Pardo ([ 9 ]) and also give some examples which show that closed and τ -closed submodules are different concepts....
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...We state Proposition 2.11 in [ 9 ] with a different proof for completeness in the following proposition....
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...We give Proposition 2.9 in [ 9 ] with a slight different proof for completeness in the following proposition and next, we give a condition for the converse of this proposition....
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...In [ 9 ], Pardo also defined τ -complement submodules as follows....
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...Motivated by the notions of these generalized extending modules, we introduce here the concept of τ -extending modules by using the concept of τ -large submodules defined by Pardo ([ 9 ]) in 1985....
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