scholarly journals On Jordan ideals and Generalized (α, 1)- Reverse derivations in ∗-prime rings

2021 ◽  
Vol 23 (11) ◽  
pp. 236-242
Author(s):  
Sk. Haseena ◽  
◽  
Chennupalle Divya ◽  
C. Jaya Subba Reddy ◽  
◽  
...  
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

Let R will be a 2- torsion free ∗-prime ring and α be an automorphisum of R. F be a nonzero generalized (α, 1)- reverse derivation of R with associated nonzero (α, 1)- reverse derivation d which commutes with ∗ and J be a nonzero ∗-Jordan ideal and a subring of R. In the present paper, we shall prove that R is commutative if any one of the following holds: (i) [F(u), u]α,1 = 0, (ii) F(u) α(u) = ud(u), (iii) F(u2) = ± α(u2), (iv) F(u2) = 2d(u) α(u), (v) d(u2) = 2F(u) α(u), for all u ∈ U.

scholarly journals On Higher N-Derivation Of Prime Rings

2014 ◽  
Vol 11 (2) ◽  
pp. 211-219
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

scholarly journals Jordan Derivations on Lie Ideals of ?-Prime Rings

2017 ◽  
Vol 36 ◽  
pp. 1-5
Author(s):  
Akhil Chandra Paul ◽  
Md Mizanor Rahman
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Lie Ideal ◽  
Prime Rings ◽  
R And D ◽  
Square Closed Lie Ideal ◽  
Torsion Free

In this paper we prove that, if U is a s-square closed Lie ideal of a 2-torsion free s-prime ring R and  d: R(R is an additive mapping satisfying d(u2)=d(u)u+ud(u) for all u?U then d(uv)=d(u)v+ud(v) holds for all  u,v?UGANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 1-5

scholarly journals Semiderivations Satisfying Certain Algebraic Identities on Jordan Ideals

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Abdellah Mamouni ◽  
Lahcen Oukhtite
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
Torsion Free

Let be a ring. An additive mapping is called semiderivation of if there exists an endomorphism of such that and , for all in . Here we prove that if is a 2-torsion free -prime ring and a nonzero -Jordan ideal of such that for all , then either is commutative or for all . Moreover, we initiate the study of generalized semiderivations in prime rings.

scholarly journals On the commutativity of semi-prime rings

1982 ◽  
Vol 32 (1) ◽  
pp. 48-51
Author(s):  
Abdullah H. Al-Moajil
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

AbstractIt is shown that ifRis a 2-torsion-free semi-prime ring such that [xy, [xy, yx]] = 0 for allx, y∈R, thenRis commutative.

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.

Generalized derivations on Lie ideals in prime rings

2016 ◽  
Vol 10 (02) ◽  
pp. 1750032 ◽  
Author(s):  
V. K. Yadav ◽  
S. K. Tiwari ◽  
R. K. Sharma
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Square Closed Lie Ideal ◽  
Torsion Free

Let [Formula: see text] be a [Formula: see text]-torsion free prime ring, and [Formula: see text] a square closed Lie ideal of [Formula: see text] Further let [Formula: see text] and [Formula: see text] be generalized derivations associated with derivations [Formula: see text] and [Formula: see text], respectively on [Formula: see text] If one of the following conditions holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] for all [Formula: see text] then it is proved that either [Formula: see text] or [Formula: see text]

scholarly journals Generalized Derivations Acting as Homomorphisms and Anti-Homomorphisms on Lie Ideals of Prime Rings

2016 ◽  
Vol 35 ◽  
pp. 73-77
Author(s):  
Akhil Chandra Paul ◽  
Sujoy Chakraborty
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Lie Ideal ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Square Closed Lie Ideal ◽  
Torsion Free

Let U be a non-zero square closed Lie ideal of a 2-torsion free prime ring R and f a generalized derivation of R with the associated derivation d of R. If f acts as a homomorphism and as an anti-homomorphism on U, then we prove that d = 0 or U € Z(R), the centre of R.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 73-77

scholarly journals Jordan higher derivations in prime ?-rings

2016 ◽  
Vol 27 (2) ◽  
pp. 155-163
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Higher Derivation ◽  
Torsion Free

The objective of this paper is to study Jordan higher derivations in prime ? -rings. We introduce a higher derivation and a Jordan higher derivation in ? -rings. For a 2-torsion free prime ? -ring M which satisfies the condition a?b?c = a?b?c for all a,b, c?M and ? ,? ?? , we prove that every Jordan higher derivation D = (di)i?N0 of M is a higher derivation of M Bangladesh J. Sci. Res. 27(2): 155-163, December-2014

scholarly journals (U, M) -derivations in prime ?-rings

2016 ◽  
Vol 27 (2) ◽  
pp. 143-153
Author(s):  
MM Rahman ◽  
AC Paul
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Prime Rings ◽  
Torsion Free

In this article, we define (U,M)-derivation d of a ? -ring M . For a Lie ideal U of a 2 - torsion free prime ? -ring M satisfying the condition a?b?c = a?b?c for all a,b, c?M and ? ,? ?? , we prove the following results:(i) ifU is an admissible Lie ideal of M, then d(u?v) = d(u)?v + u?d(v) for all u, v?U ,? ??(ii) if u?u?U for all u?U,? ?? , then d(u?m) = d(u)?m + u?d(m) for all m ? M Bangladesh J. Sci. Res. 27(2): 143-153, December-2014

scholarly journals A Generalized Higher Reverse Left (accordingly, Right) Centralizer on Prime Rings

2019 ◽  
Vol 54 (5) ◽  
Author(s):  
Fawaz Ra’ad Jarullah ◽  
Salah Mehdi Salih
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Definition Of ◽  
Torsion Free

In this study we introduced the concepts of generalized higher reverse left (accordingly, right) centralizer, and Jordan generalized higher reverse left (accordingly, right) centralizer of rings. The definition of Jordan triple generalized higher reverse left (accordingly, right) centralizer was deduced. The most important findings of this paper are as follows: every Jordan generalized higher reverse left (accordingly right) centralizer of a 2-torsion free prime ring R into itself is a generalized higher reverse left (accordingly right) centralizer of R. The results have confirmed that every Jordan generalized higher reverse left (accordingly right) centralizer is a generalized higher reverse left (accordingly right) centralizer within certain conditions.

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