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Journal ArticleDOI

Functional identities of degree 2 in triangular rings revisited

04 Mar 2015-Linear & Multilinear Algebra (Taylor & Francis)-Vol. 63, Iss: 3, pp 534-553
TL;DR: Using the maximal left ring of quotients, the authors generalized the result on functional identity in triangular rings to the case of functional identity on commuting additive maps and generalized inner biderivations of triangular rings.
Abstract: Using the notion of the maximal left ring of quotients, our recent result on the solutions of functional identity in triangular rings is generalized. Consequently, generalizations of known results on commuting additive maps and generalized inner biderivations of triangular rings are obtained.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors give the solutions of functional identities of degree 2 in arbitrary triangular rings, which generalizes a recent result due to Eremita, for commuting additive maps and generalized inner biderivations.

16 citations

Journal ArticleDOI
TL;DR: Benkovic and Ghosseiri as mentioned in this paper proved that every biderivation of certain triangular rings is the sum of an extremal biderization and an inner biderisation, using the notion of maximal left ring of quotients.
Abstract: In this paper, we prove that every biderivation of certain triangular rings is the sum of an extremal biderivation and an inner biderivation, using the notion of maximal left ring of quotients. As a consequence, we show that every biderivation of the ring of all upper triangular matrices over a unital ring () is the sum of an extremal biderivation and an inner biderivation, which extends two results of Benkovic and Ghosseiri, respectively.

9 citations


Cites background from "Functional identities of degree 2 i..."

  • ...Recently, Eremita [13] investigated functional identities of degree 2 for a much wide class of triangular rings, using the notion of maximal left ring of quotients....

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  • ...In 2013, Eremita [12] initiated the study of functional identities of degree 2 in certain triangular rings....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that if R is a d-free subset of a ring Q, then the upper triangular matrix ring T n (R ) is also a d free subset of Q for any n ∈ N.

7 citations

Journal ArticleDOI
TL;DR: In this paper, a generalized matrix algebra over a commutative ring and the center of the trace of a -bilinear mapping is considered, and sufficient conditions for each Lie triple isomorphism of such a trace are established.
Abstract: Let be a generalized matrix algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and is the trace of . We describe the form of satisfying the condition for all . The question of when has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

7 citations

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.

1,584 citations

Book
17 Nov 1995
TL;DR: Preliminaries ring of quotients orthogonal completions primitive rings with GPI T-identities of prime rings and semiprime rings applications of Lie theory as mentioned in this paper.
Abstract: Preliminaries ring of quotients orthogonal completions primitive rings the PBW theorem rings with GPI T-identities of prime rings T-identities of semiprime rings applications of Lie theory.

716 citations

Journal ArticleDOI

370 citations


"Functional identities of degree 2 i..." refers background in this paper

  • ...In the beginning of 1990s Brešar described the form of commuting additive maps [3] and commuting traces of biadditive maps, and in [4] he also considered functional identity (1....

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Journal ArticleDOI
TL;DR: In this paper, it was shown that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0.
Abstract: Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.

286 citations

Journal ArticleDOI
TL;DR: In this article, the authors survey the development of the theory of commuting maps and their applications, including derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to Lie theory.
Abstract: A map $f$ on a ring $\cal A$ is said to be commuting if $f(x)$ commutes with $x$ for every $x\in \cal A$. The paper surveys the development of the theory of commuting maps and their applications. The following topics are discussed: commuting derivations, commuting additive maps, commuting traces of multiadditive maps, various generalizations of the notion of a commuting map, and applications of results on commuting maps to different areas, in particular to Lie theory.

193 citations


"Functional identities of degree 2 i..." refers background in this paper

  • ...Commuting maps appear in many areas and have been studied intensively (see the survey article [1] or [2, p....

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