scholarly journals Nonlinear generalized Jordan (σ, Γ)-derivations on triangular algebras

2017 ◽  
Vol 6 (1) ◽  
pp. 216-228
Author(s):  
Ahmad N. Alkenani ◽  
Mohammad Ashraf ◽  
Aisha Jabeen
Keyword(s):  
Commutative Ring ◽  
Identity Element ◽  
Triangular Algebra ◽  
Triangular Algebras ◽  
Torsion Free

Abstract Let R be a commutative ring with identity element, A and B be unital algebras over R and let M be (A,B)-bimodule which is faithful as a left A-module and also faithful as a right B-module. Suppose that A = Tri(A,M,B) is a triangular algebra which is 2-torsion free and σ, Γ be automorphisms of A. A map δ:A→A (not necessarily linear) is called a multiplicative generalized (σ, Γ)-derivation (resp. multiplicative generalized Jordan (σ, Γ)-derivation) on A associated with a (σ, Γ)-derivation (resp. Jordan (σ, Γ)-derivation) d on A if δ(xy) = δ(x)r(y) + σ(x)d(y) (resp. σ(x<sup>2</sup>) = δ(x)r(x) + δ(x)d(x)) holds for all x, y Є A. In the present paper it is shown that if δ:A→A is a multiplicative generalized Jordan (σ, Γ)-derivation on A, then δ is an additive generalized (σ, Γ)-derivation on A.

scholarly journals Generalized Higher Derivations on Lie Ideals of Triangular Algebras

2017 ◽  
Vol 25 (1) ◽  
pp. 35-53
Author(s):  
Mohammad Ashraf ◽  
Nazia Parveen ◽  
Bilal Ahmad Wani
Keyword(s):  
Commutative Ring ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Square Closed Lie Ideal ◽  
Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

Nonlinear Lie triple derivations by local actions on triangular algebras

2021 ◽  
Author(s):  
Xingpeng Zhao
Keyword(s):  
Commutative Ring ◽  
Triangular Algebra ◽  
Mild Conditions ◽  
Derivation Formula ◽  
Triangular Algebras

Let [Formula: see text] be a triangular algebra over a commutative ring [Formula: see text]. In this paper, under some mild conditions on [Formula: see text], we prove that if [Formula: see text] is a nonlinear map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text]. Then [Formula: see text] is almost additive on [Formula: see text], that is, [Formula: see text] Moreover, there exist an additive derivation [Formula: see text] of [Formula: see text] and a nonlinear map [Formula: see text] such that [Formula: see text] for [Formula: see text], where [Formula: see text] for any [Formula: see text] with [Formula: see text].

scholarly journals Generalized Jordan triple (σ,τ)-higher derivation on triangular algebras

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2285-2294
Author(s):  
Mohammad Ashraf ◽  
Aisha Jabeen ◽  
Mohd Akhtar
Keyword(s):  
Commutative Ring ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Triangular Algebras

Let R be a commutative ring with unity, U = Tri(A,M,B) be a triangular algebra consisting of unital algebras A,B and (A,B)-bimodule M which is faithful as a left A-module and also as a right B-module. Let ? and ? be two automorphisms of U. A family ? = {?n}n?N of R-linear mappings ?n : U ? U is said to be a generalized Jordan triple (?,?)-higher derivation on A if there exists a Jordan triple (?,?)-higher derivation D = {dn}n?N on U such that ?0 = IU, the identity map of U and ?n(XYX) = ?i+j+k=n ?i(?n-i(X))dj(?k?i(Y))dk(?n-k(X)) holds for all X,Y ? U and each n ? N. In this article, we study generalized Jordan triple (?,?)-higher derivation on A and prove that every generalized Jordan triple (?,?)-higher derivation on U is a generalized (?,?)-higher derivation on U.

scholarly journals A Class of Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xiuhai Fei ◽  
Haifang Zhang
Keyword(s):  
Matrix Algebra ◽  
Triangular Matrix ◽  
Nest Algebra ◽  
Triangular Algebra ◽  
Triangular Matrix Algebra ◽  
Free Block ◽  
Upper Triangular Matrix ◽  
Triangular Algebras ◽  
Torsion Free

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

scholarly journals Zero-Sum Coefficient Derivations in Three Variables of Triangular Algebras

2016 ◽  
Vol 8 (5) ◽  
pp. 37
Author(s):  
Youngsoo Kim ◽  
Byunghoon Lee
Keyword(s):  
Standard Form ◽  
Sufficient Conditions ◽  
Multilinear Polynomial ◽  
Triangular Algebra ◽  
Triangular Algebras ◽  
Zero Sum ◽  
Lie Derivations

Under mild assumptions Benkovi\v{c} showed that an $f$-derivation of a triangular algebra is a derivation when the sum of the coefficients of the multilinear polynomial $f$ is nonzero. We investigate the structure of $f$-derivations of triangular algebras when $f$ is of degree 3 and the coefficient sum is zero. The zero-sum coeffient derivations include Lie derivations (degree 2) and Lie triple derivations (degree 3), which have been previously shown to be not necessarily derivations but in standard form, i.e., the sum of a derivation and a central map. In this paper, we present sufficient conditions on the coefficients of $f$ to ensure that any $f$-derivations are derivations or are in standard form.<br /><br />

scholarly journals On the Quadratic Extensions and the Extended Witt Ring of a Commutative Ring

1973 ◽  
Vol 49 ◽  
pp. 127-141 ◽  
Author(s):  
Teruo Kanzaki
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Identity Element ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Witt Ring ◽  
Division Rings ◽  
Quadratic Extensions ◽  
The Common

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

Simple reduction theorems for extension and torsion functors

1961 ◽  
Vol 57 (3) ◽  
pp. 483-488
Author(s):  
D. G. Northcott
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Residue Class ◽  
Identity Element ◽  
Negative Integer ◽  
Reduction Theorem ◽  
Residue Class Ring ◽  
Image Position ◽  
Class Ring ◽  
Simple Reduction

Let Λ be a commutative ring with an identity element and c an element of Λ which is not a zero divisor Denote by Ω the residue class ring Λ/Λc. If now M is a Λ-module for which c is not a zero divisor, and A is an Ω-module, then a theorem of Rees (2) asserts that, for every non-negative integer n, we have a Λ-isomorphismThis reduction theorem has found a number of useful and interesting applications.

scholarly journals Rings all of whose torsion quasi-injective modules are injective

1984 ◽  
Vol 25 (2) ◽  
pp. 219-227 ◽  
Author(s):  
J. Ahsan ◽  
E. Enochs
Keyword(s):  
Jacobson Radical ◽  
Identity Element ◽  
Torsion Theory ◽  
Injective Hull ◽  
Torsion Class ◽  
Injective Modules ◽  
Torsion Theories ◽  
Torsion Free

Throughout this paper it is assumed that rings are associative, have the identity element, and all modules are left unital. R will denote a ring with identity, R-Mod the category of left R-modules, and for each left R-module M, E(M) (resp. J(M)) will represent the injective hull (resp. Jacobson radical) of M. Also, for a module M, A ⊆' M will mean that A is an essential submodule of M, and Z(M) denotes the singular submodule of M. M is called singular if Z(M) = M, and it is called non-singular in case Z(M) = 0. For fundamental definitions and results related to torsion theories, we refer to [12] and [14]. In this paper we shall deal mainly with Goldie torsion theory. Recall that a pair (G, F) of classes of left R-modules is known as Goldie torsion theory if G is the smallest torsion class containing all modules B/A, where A ⊆' B, and the torsion free class F is precisely the class of non-singular modules.

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

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