A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

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.

scholarly journals Remarks on Generalized Derivations in Prime and Semiprime Rings

2010 ◽  
Vol 2010 ◽  
pp. 1-6 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

LetRbe a ring with centerZandIa nonzero ideal ofR. An additive mappingF:R→Ris called a generalized derivation ofRif there exists a derivationd:R→Rsuch thatF(xy)=F(x)y+xd(y)for allx,y∈R. In the present paper, we prove that ifF([x,y])=±[x,y]for allx,y∈IorF(x∘y)=±(x∘y)for allx,y∈I, then the semiprime ringRmust contains a nonzero central ideal, providedd(I)≠0. In caseRis prime ring,Rmust be commutative, providedd≠0. The cases (i)F([x,y])±[x,y]∈Zand (ii)F(x∘y)±(x∘y)∈Zfor allx,y∈Iare also studied.

scholarly journals Generalized derivations in prime and semiprime

2015 ◽  
Vol 34 (2) ◽  
pp. 29
Author(s):  
Shuliang Huang ◽  
Nadeem Ur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Positive Integers

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$  fixed positive integers.  If $R$ admits a generalized derivation $F$ associated with a  nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for  all $x,y\in I$, then $R$ is commutative. Moreover  we also examine the case when $R$ is a semiprime ring.

On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

2019 ◽  
Vol 26 (01) ◽  
pp. 93-104
Author(s):  
Vincenzo De Filippis ◽  
Nadeem ur Rehman
Keyword(s):  
Prime Ring ◽  
Lie Ideal ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings.

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) .

On Automorphisms and Commutativity in Semiprime Rings

2013 ◽  
Vol 56 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Pao-Kuei Liau ◽  
Cheng-Kai Liu
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Positive Integers

Abstract. Let R be a semiprime ring with center Z(R). For x, y ∊ R, we denote by [x, y] = xy – yx the commutator of x and y. If σ is a non-identity automorphism of R such thatfor all x ∊ R, where n0, n1, n2, … nk are fixed positive integers, then there exists a map μ: R → Z(R) such that σ(x) = x + μ(x) for all x ∊ R. In particular, when R is a prime ring, R is commutative.

scholarly journals Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

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.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

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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