Centrally essential rings which are not necessarily unital or associative

2019 ◽  
Vol 29 (4) ◽  
pp. 215-218 ◽  
Author(s):  
Viktor T. Markov ◽  
Askar A. Tuganbaev
Keyword(s):  
Alternative Ring ◽  
Semiprime Rings ◽  
Unital Rings

Abstract Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.

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.

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.

scholarly journals Some results concerning n-σ-centralizing mappings in semiprime rings

2014 ◽  
Vol 3 (1) ◽  
pp. 15-21
Author(s):  
Vincenzo De Filippis ◽  
Basudeb Dhara
Keyword(s):  
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

Commutative center = center in a prime finitely generated right alternative ring

1997 ◽  
Vol 25 (10) ◽  
pp. 3147-3153 ◽  
Author(s):  
Irvin R. Hentzel ◽  
Erwin Kleinfeld ◽  
Harry F. Smith
Keyword(s):  
Finitely Generated ◽  
Alternative Ring ◽  
Right Alternative Ring
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