On centralizers of semiprime rings with involution

2006 ◽  
Vol 43 (1) ◽  
pp. 61-67 ◽  
Author(s):  
Joso Vukman ◽  
Irena Kosi-Ulbl
Keyword(s):  
Semiprime Ring ◽  
Additive Mapping ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

Let Rbe a 2-torsion free semiprime *-ring and let T:R?Rbe an additive mapping such that T(xx*)=T(x)x* is fulfilled for all x ?R. In this case T(xy)=T(x)yholds for all pairs x,y?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.

Ad-nilpotent Elements of Semiprime Rings with Involution

2018 ◽  
Vol 61 (2) ◽  
pp. 318-327
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Semiprime Ring ◽  
Extended Centroid ◽  
Nilpotent Elements ◽  
Semiprime Rings ◽  
Rings With Involution ◽  
Torsion Free

AbstractLet R be an n!-torsion free semiprime ring with involution * and with extended centroid C, where n > 1 is a positive integer. We characterize a ∊ K, the Lie algebra of skew elements in R, satisfying (ada)n = 0 on K. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if a, b ∊ R satisfy (ada)n = adb on R, where either n is even or b = 0, then (a − λ)[(n+1)/2] = 0 for some λ ∊ C.

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

scholarly journals On Jordan Triple α-*Centralizers Of Semiprime Rings

2014 ◽  
Vol 47 (1) ◽  
Author(s):  
Mohammad Ashraf ◽  
Muzibur Rahman Mozumder ◽  
Almas Khan
Keyword(s):  
Semiprime Ring ◽  
Additive Mapping ◽  
Semiprime Rings ◽  
Left And Right ◽  
Torsion Free

AbstractLet R be a 2-torsion free semiprime ring equipped with an involution *. An additive mapping T : R→R is called a left (resp. right) Jordan α-*centralizer associated with a function α: R→R if T(x2)=T(x)α(x*) (resp. T(x2)=α(x*)T(x)) holds for all x ∊ R. If T is both left and right Jordan α-* centralizer of R, then it is called Jordan α-* centralizer of R. In the present paper it is shown that if α is an automorphism of R, and T : R→ R is an additive mapping such that 2T(xyx)=T(x) α(y*x*) +α(x*y*)T(x) holds for all x, y ∊ R, then T is a Jordan α-*centralizer of R

On Certain Functional Equation in Semiprime Rings

2016 ◽  
Vol 23 (01) ◽  
pp. 65-70
Author(s):  
Nejc Širovnik ◽  
Joso Vukman
Keyword(s):  
Functional Equation ◽  
Semiprime Ring ◽  
Identity Element ◽  
Additive Mapping ◽  
Fixed Integer ◽  
Fixed Element ◽  
Semiprime Rings ◽  
Torsion Free

Let n be a fixed integer, let R be an (n+1)!-torsion free semiprime ring with the identity element and let F: R → R be an additive mapping satisfying the relation [Formula: see text] for all x ∈ R. In this case, we prove that F is of the form 2F(x)=D(x)+ax+xa for all x ∈ R, where D: R → R is a derivation and a ∈ R is some fixed element.

scholarly journals Jordan ?-Centralizers of Prime and Semiprime Rings

2010 ◽  
Vol 7 (4) ◽  
pp. 1426-1431
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Zero Divisor ◽  
Semiprime Ring ◽  
Additive Mapping ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Torsion Free

The purpose of this paper is to prove the following result: Let R be a 2-torsion free ring and T: R?R an additive mapping such that T is left (right) Jordan ?-centralizers on R. Then T is a left (right) ?-centralizer of R, if one of the following conditions hold (i) R is a semiprime ring has a commutator which is not a zero divisor . (ii) R is a non commutative prime ring . (iii) R is a commutative semiprime ring, where ? be surjective endomorphism of R . It is also proved that if T(x?y)=T(x)??(y)=?(x)?T(y) for all x, y ? R and ?-centralizers of R coincide under same condition and ?(Z(R)) = Z(R) .

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.

A NOTE ON -JORDAN DERIVATIONS OF RINGS AND BANACH ALGEBRAS

2015 ◽  
Vol 93 (2) ◽  
pp. 231-237 ◽  
Author(s):  
IRENA KOSI-ULBL ◽  
JOSO VUKMAN
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Torsion Free

In this paper we prove the following result: let$m,n\geq 1$be distinct integers, let$R$be an$mn(m+n)|m-n|$-torsion free semiprime ring and let$D:R\rightarrow R$be an$(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation$(m+n)D(x^{2})=2mD(x)x+2nxD(x)$for$x\in R$. Then$D$is a derivation which maps$R$into its centre.

Commutativity Conditions on Rings with Involution

1982 ◽  
Vol 34 (1) ◽  
pp. 17-22
Author(s):  
Paola Misso
Keyword(s):  
Semiprime Ring ◽  
Central Elements ◽  
Rings With Involution ◽  
Image Position ◽  
Partial Extension ◽  
Torsion Free

Let R be a ring with involution *. We denote by S, K and Z = Z(R) the symmetric, the skew and the central elements of R respectively.In [4] Herstein defined the hypercenter T(R) of a ring R asand he proved that in case R is without non-zero nil ideals then T(R) = Z(R).In this paper we offer a partial extension of this result to rings with involution.We focus our attention on the following subring of R:(We shall write H(R) as H whenever there is no confusion as to the ring in question.)Clearly H contains the central elements of R. Our aim is to show that in a semiprime ring R with involution which is 2 and 3-torsion free, the symmetric elements of H are central.

scholarly journals On Generalized ()-Derivations in Semiprime Rings

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Basudeb Dhara ◽  
Atanu Pattanayak
Keyword(s):  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Semiprime Rings

Let be a semiprime ring, a nonzero ideal of , and , two epimorphisms of . An additive mapping is generalized -derivation on if there exists a -derivation such that holds for all . In this paper, it is shown that if , then contains a nonzero central ideal of , if one of the following holds: (i) ; (ii) ; (iii) ; (iv) ; (v) for all .

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