Every PI-ring Satisfies a Power of the Standard Identity

2004 ◽  
pp. 161-162
Author(s):  
Vesselin Drensky ◽  
Edward Formanek
Keyword(s):  
Standard Identity

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.

scholarly journals VOROS PRODUCT AND NONCOMMUTATIVE INSPIRED BLACK HOLES

2013 ◽  
Vol 28 (09) ◽  
pp. 1350030
Author(s):  
SUNANDAN GANGOPADHYAY
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Extremal Point ◽  
De Sitter ◽  
Schwarzschild Black Hole ◽  
Standard Identity ◽  
Schwarzschild Geometry ◽  
Area Law

We emphasize the importance of the Voros product in defining the noncommutative (NC) inspired black holes. The computation of entropy for both the noncommutative inspired Schwarzschild and Reissner–Nordström (RN) black holes show that the area law holds up to order [Formula: see text]. The leading correction to the entropy (computed in the tunneling formalism) is shown to be logarithmic. The Komar energy E for these black holes is then obtained and a deviation from the standard identity E = 2STH is found at the order [Formula: see text]. This deviation leads to a nonvanishing Komar energy at the extremal point TH = 0 of these black holes. The Smarr formula is finally worked out for the NC Schwarzschild black hole. Similar features also exist for a de Sitter–Schwarzschild geometry.

Actions of Generalized Derivations on Multilinear Polynomials in Prime Rings

2011 ◽  
Vol 18 (spec01) ◽  
pp. 955-964 ◽  
Author(s):  
Nurcan Argaç ◽  
Vincenzo De Filippis
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Utumi Quotient Ring ◽  
Standard Identity ◽  
Multilinear Polynomials

Let K be a commutative ring with unity, R a non-commutative prime K-algebra with center Z(R), U the Utumi quotient ring of R, C=Z(U) the extended centroid of R, I a non-zero two-sided ideal of R, H and G non-zero generalized derivations of R. Suppose that f(x1,…,xn) is a non-central multilinear polynomial over K such that H(f(X))f(X)-f(X)G(f(X))=0 for all X=(x1,…,xn)∈ In. Then one of the following holds: (1) There exists a ∈ U such that H(x)=xa and G(x)=ax for all x ∈ R. (2) f(x1,…,xn)2 is central valued on R and there exist a, b ∈ U such that H(x)=ax+xb and G(x)=bx+xa for all x ∈ R. (3) char (R)=2 and R satisfies s4, the standard identity of degree 4.

scholarly journals The minimal degree standard identity on MnE2 and MnE3

2020 ◽  
Vol 238 (1) ◽  
pp. 279-312
Author(s):  
Barbara Anna Balázs ◽  
Szabolcs Mészáros
Keyword(s):  
Minimal Degree ◽  
Standard Identity

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.

Generalized Derivations with Power-Central Values on Multilinear Polynomials

2006 ◽  
Vol 13 (03) ◽  
pp. 405-410 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Commutative Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Algebra ◽  
Standard Identity ◽  
Central Values ◽  
Multilinear Polynomials ◽  
Power Central

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X1, …,Xt) a multilinear polynomial over K. If g(f (X1, …,Xt))n ∈ Z for all x1, …, xt ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x1, …,xt) is central-valued on R except when R satisfies s4, the standard identity in four variables.

The minimal degree standard identity on E ⊗ E

2021 ◽  
Vol 15 (4) ◽  
pp. 277-281
Author(s):  
Geraldo de Assis Junior ◽  
Sergio Mota Alves
Keyword(s):  
Minimal Degree ◽  
Standard Identity

Remark on the standard identity

1978 ◽  
Vol 23 (5) ◽  
pp. 414-416 ◽  
Author(s):  
A. R. Kemer
Keyword(s):  
Standard Identity
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