scholarly journals A Note on Jordan Triple Higher *-Derivations on Semiprime Rings

ISRN Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
O. H. Ezzat
Keyword(s):  
Semiprime Ring ◽  
Semiprime Rings ◽  
Additive Mappings ◽  
Torsion Free

We introduce the following notion. Let ℕ0 be the set of all nonnegative integers and let D=(di)i∈ℕ0 be a family of additive mappings of a *-ring R such that d0=idR; D is called a Jordan higher *-derivation (resp., a Jordan higher *-derivation) of R if dn(x2)=∑i+j=n‍di(x)dj(x*i) (resp., dn(xyx)=∑i+j+k=n‍di(x)dj(y*i)dk(x*i+j)) for all x,y∈R and each n∈ℕ0. It is shown that the notions of Jordan higher *-derivations and Jordan triple higher *-derivations on a 6-torsion free semiprime *-ring are coincident.

A Result Concerning Additive Mappings in Semiprime Rings

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Maja Fos̆ner
Keyword(s):  
Semiprime Ring ◽  
Additive Mapping ◽  
Semiprime Rings ◽  
Additive Mappings ◽  
Torsion Free

AbstractIn this paper we prove the following result. Let R be a 2-torsion free semiprime ring and let f : R → R be an additive mapping satisfying the relation f(x)x

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.

On commuting additive mappings on semiprime rings

2020 ◽  
pp. 2150079
Author(s):  
Siriporn Lapuangkham ◽  
Utsanee Leerawat
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Semiprime Rings ◽  
Additive Mappings

The main purpose of this paper is to describe the structure of a pair of additive mappings that are commuting on a semiprime ring. Furthermore, we prove that the existence of different commuting epimorphisms on a prime ring forces the ring to be commutative. Finally, we characterize additive mappings, which act as homomorphisms or antihomomorphisms on a semiprime ring.

scholarly journals Jordan derivations on 2-Torsion free semiprime ?-Rings

2014 ◽  
Vol 38 (2) ◽  
pp. 189-195
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Semiprime Ring ◽  
Jordan Derivation ◽  
Semiprime Rings ◽  
Academy Of Sciences ◽  
Torsion Free

The objective of this paper was to study Jordan derivations on semiprime ?-ring. Let M be a 2-torsion free semiprime ?-ring satisfying the condition a?b?c = a?b?c for all a,b,c ? M and ?, ? ? ?. The authors proved that every Jordan derivation of M is a derivation of M. DOI: http://dx.doi.org/10.3329/jbas.v38i2.21343 Journal of Bangladesh Academy of Sciences, Vol. 38, No. 2, 189-195, 2014

scholarly journals On symmetric biadditive mappings of semiprime rings

2017 ◽  
Vol 35 (1) ◽  
pp. 9
Author(s):  
Asma Ali ◽  
Khalid Ali Hamdin ◽  
Shahoor Khan
Keyword(s):  
Semiprime Ring ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring with centre Z(R). A mapping D(., .) : R× R −→ R issaid to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ Rdefined by f(x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious thatin the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive(i.e. additive in both arguments), the trace f of D satisfies the relation f(x + y) =f(x) + f(y) + 2D(x, y), for all x, y ∈ R. In this paper we prove that a nonzero left idealL of a 2-torsion free semiprime ring R is central if it satisfies any one of the followingproperties: (i) f(xy) ∓ [x, y] ∈ Z(R), (ii) f(xy) ∓ [y, x] ∈ Z(R), (iii) f(xy) ∓ xy ∈Z(R), (iv) f(xy)∓yx ∈ Z(R), (v) f([x, y])∓[x, y] ∈ Z(R), (vi) f([x, y])∓[y, x] ∈ Z(R),(vii) f([x, y])∓xy ∈ Z(R), (viii) f([x, y])∓yx ∈ Z(R), (ix) f(xy)∓f(x)∓[x, y] ∈ Z(R),(x) f(xy)∓f(y)∓[x, y] ∈ Z(R), (xi) f([x, y])∓f(x)∓[x, y] ∈ Z(R), (xii) f([x, y])∓f(y)∓[x, y] ∈ Z(R), (xiii) f([x, y])∓f(xy)∓[x, y] ∈ Z(R), (xiv) f([x, y])∓f(xy)∓[y, x] ∈ Z(R),(xv) f(x)f(y) ∓ [x, y] ∈ Z(R), (xvi) f(x)f(y) ∓ [y, x] ∈ Z(R), (xvii) f(x)f(y) ∓ xy ∈Z(R), (xviii) f(x)f(y) ∓ yx ∈ Z(R), (xix) f(x) ◦ f(y) ∓ [x, y] ∈ Z(R), (xx) f(x) ◦f(y) ∓ xy ∈ Z(R), (xxi) f(x) ◦ f(y) ∓ yx ∈ Z(R), (xxii) f(x)f(y) ∓ x ◦ y ∈ Z(R),(xxiii) [x, y] − f(xy) + f(yx) ∈ Z(R), for all x, y ∈ R, where f stands for the trace of asymmetric biadditive mapping D(., .) : R × R −→ R.

On generalized (θ, φ)-derivations in semiprime rings with involution

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Mohammad Ashraf ◽  
Nadeem-ur-Rehman ◽  
Shakir Ali ◽  
Muzibur Mozumder
Keyword(s):  
Semiprime Ring ◽  
Additive Mapping ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

AbstractThe main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

scholarly journals Derivations of higher order in semiprime rings

1998 ◽  
Vol 21 (1) ◽  
pp. 89-92 ◽  
Author(s):  
Jiang Luh ◽  
Youpei Ye
Keyword(s):  
Positive Integer ◽  
Semiprime Ring ◽  
Higher Order ◽  
Inner Derivation ◽  
Semiprime Rings ◽  
Torsion Free

LetRbe a2-torsion free semiprime ring with derivationd. Supposedd2nis a derivation ofR, wherenis a positive integer. It is shown that ifRis(4n−2)-torsion free or ifRis an inner derivation ofR, thend2n−1=0.

On Prime and Semiprime Rings with Derivations

2006 ◽  
Vol 13 (03) ◽  
pp. 371-380 ◽  
Author(s):  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Nonempty Subset ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

ON n-CENTRALIZING GENERALIZED DERIVATIONS IN SEMIPRIME RINGS WITH APPLICATIONS TO C*-ALGEBRAS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250111 ◽  
Author(s):  
BASUDEB DHARA ◽  
SHAKIR ALI
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Generalized Derivations ◽  
C Algebras ◽  
Semiprime Rings ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xn – xn f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

scholarly journals (ϕ, φ)-derivations on semiprime rings and Banach algebras

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bilal Ahmad Wani
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Additive Mappings

Abstract Let ℛ be a semiprime ring with unity e and ϕ, φ be automorphisms of ℛ. In this paper it is shown that if ℛ satisfies 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 is an (ϕ, φ)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, Gscr; : ℛ → ℛ satisfying the relations 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒢 ( x ) + 𝒢 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒢 ( x n - 1 ) , 2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{G}\left( x \right) + \mathcal{G}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{G}\left( {{x^{n - 1}}} \right), 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) D ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) , 2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right), for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 and 𝒢 are (ϕ, φ)- derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

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