(δ, ε)-Differential Identities of UT m(F)

2020 ◽  
pp. 319-333
Author(s):  
Vincenzo C. Nardozza
Keyword(s):  
Differential Identities

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.

Some results on ∗-differential identities in prime rings

2021 ◽  
Author(s):  
Deepak Kumar ◽  
Bharat Bhushan ◽  
Gurninder S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Derivation Formula

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

scholarly journals DIFFICULTIES OF AN INFRARED EXTENSION OF DIFFERENTIAL RENORMALIZATION

1994 ◽  
Vol 09 (07) ◽  
pp. 1067-1096 ◽  
Author(s):  
L. V. AVDEEV ◽  
D. I. KAZAKOV ◽  
I. N. KONDRASHUK
Keyword(s):  
Perturbation Theory ◽  
Fourier Transformation ◽  
Two Dimensions ◽  
Test Scheme ◽  
Two Dimensional ◽  
Differential Identities ◽  
Feynman Integrals ◽  
Differential Renormalization ◽  
Self Consistent ◽  
Infrared Singularities

We investigate the possibility of generalizing the differential renormalization of D. Z. Freedman, K. Johnson and J. I. Latorre in an invariant fashion to theories with infrared divergencies via an infrared [Formula: see text] operation. Two-dimensional σ models and the four-dimensional ɸ4-theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. A consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on the infrared extension of differential renormalization lead to incorrect results. We conclude that the procedure of extended differential renormalization does not perform the infrared [Formula: see text] operation in a self-consistent way.

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