Some differential identities on prime and semiprime rings and Banach algebras

2018 ◽  
Vol 68 (2) ◽  
pp. 305-313 ◽  
Author(s):  
Mohd Arif Raza ◽  
Mohammad Shadab Khan ◽  
Nadeem ur Rehman
Keyword(s):  
Banach Algebras ◽  
Differential Identities ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

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 derivations in prime and semiprime rings with Banach algebras

2018 ◽  
Vol 12 (8) ◽  
pp. 297-309
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeem ur Rehman
Keyword(s):  
Banach Algebras ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Annihilator condition of a pair of derivations in prime and semiprime rings

2016 ◽  
Vol 47 (1) ◽  
pp. 111-124 ◽  
Author(s):  
Basudeb Dhara ◽  
Nurcan Argaç ◽  
Krishna Gopal Pradhan
Keyword(s):  
Prime And Semiprime Rings ◽  
Semiprime Rings

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.

Semi-Prime Modules

1966 ◽  
Vol 18 ◽  
pp. 823-831 ◽  
Author(s):  
E. H. Feller ◽  
E. W. Swokowski
Keyword(s):  
Prime Rings ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

scholarly journals A note on a pair of derivations of semiprime rings

2004 ◽  
Vol 2004 (39) ◽  
pp. 2097-2102 ◽  
Author(s):  
Muhammad Anwar Chaudhry ◽  
A. B. Thaheem
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Semiprime Rings

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: letRbe a semiprime ring with centerZ(R), and letf,gbe derivations ofRsuch thatf(x)x+xg(x)∈Z(R)for allx∈R, thenfandgare central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivationfof a semiprime ringRis skew-commuting.

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