scholarly journals Commutativity of $\Gamma$-Generalized Boolean Semirings with Derivations

2016 ◽  
Vol 8 (4) ◽  
pp. 132 ◽  
Author(s):  
Tossatham Makkala ◽  
Utsanee Leerawat
Keyword(s):  
Generalized Derivation ◽  
Boolean Semiring ◽  
Boolean Semirings

In this paper the notion of derivations on $\Gamma$-generalized Boolean semiring are established, namely $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation. We also investigate the commutativity of prime $\Gamma$-generalized Boolean semiring admitting $\Gamma$-$(f, g)$ derivation and $\Gamma$-$(f, g)$ generalized derivation satisfying some conditions.

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.

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.

On continuous linear generalized derivation in rings and Banach algebras

2019 ◽  
pp. 2150020
Author(s):  
Nadeem ur Rehman
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Generalized Derivation ◽  
Continuous Linear ◽  
Generalized Derivations ◽  
Derivation Formula

In this paper, we investigate the commutativity of a prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivation [Formula: see text] associated with continuous linear derivation [Formula: see text] such that either [Formula: see text] or [Formula: see text] for intergers [Formula: see text] and [Formula: see text] and sufficiently many [Formula: see text]. Further, similar results are also obtained for unital prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivations [Formula: see text] satisfying either [Formula: see text] or [Formula: see text] for an integer [Formula: see text] and sufficiently many [Formula: see text].

scholarly journals Some identities involving multiplicative generalized derivations in orime and semiprime rings

2018 ◽  
Vol 36 (1) ◽  
pp. 25 ◽  
Author(s):  
Basudeb Dhara
Keyword(s):  
Generalized Derivation ◽  
Generalized Derivations ◽  
Semiprime Rings

Let $R$ be a ring with center $Z(R)$. A mapping $F:R\rightarrow R$ is called a multiplicative generalized derivation, if $F(xy)=F(x)y+xg(y)$ is fulfilled for all $x,y\in R$, where $g:R\rightarrow R$ is a derivation. In the present paper, our main object is to study the situations: (1) $F(xy)- F(x)F(y)\in Z(R)$, (2) $F(xy)+ F(x)F(y)\in Z(R)$, (3) $F(xy)- F(y)F(x)\in Z(R)$, (4) $F(xy)+ F(y)F(x)\in Z(R)$, (5) $F(xy)- g(y)F(x)\in Z(R)$; for all $x,y$ in some suitable subset of $R$.

scholarly journals Wavelength dependence of the linear growth rate of the <I>E<sub>s</sub></I> layer instability

2007 ◽  
Vol 25 (6) ◽  
pp. 1311-1322 ◽  
Author(s):  
R. B. Cosgrove
Keyword(s):  
Growth Rate ◽  
Electric Fields ◽  
Large Scale ◽  
Linear Growth ◽  
Three Dimensional ◽  
Generalized Derivation ◽  
Wavelength Dependence ◽  
Meridional Wind ◽  
Linear Growth Rate ◽  
Significant Wavelength

Abstract. It has recently been shown, by computation of the linear growth rate, that midlatitude sporadic-E (Es) layers are subject to a large scale electrodynamic instability. This instability is a logical candidate to explain certain frontal structuring events, and polarization electric fields, which have been observed in Es layers by ionosondes, by coherent scatter radars, and by rockets. However, the original growth rate derivation assumed an infinitely thin Es layer, and therefore did not address the short wavelength cutoff. Also, the same derivation ignored the effects of F region loading, which is a significant wavelength dependent effect. Herein is given a generalized derivation that remedies both these short comings, and thereby allows a computation of the wavelength dependence of the linear growth rate, as well as computations of various threshold conditions. The wavelength dependence of the linear growth rate is compared with observed periodicities, and the role of the zeroth order meridional wind is explored. A three-dimensional paper model is used to explain the instability geometry, which has been defined formally in previous works.

Derivations vanishing on commutators with generalized derivation of order 2 in prime rings

2016 ◽  
Vol 45 (8) ◽  
pp. 3542-3554 ◽  
Author(s):  
S. K. Tiwari ◽  
R. K. Sharma ◽  
B. Dhara
Keyword(s):  
Generalized Derivation ◽  
Prime Rings

On Some Generalized Identities with Derivations on Multilinear Polynomials

2010 ◽  
Vol 17 (02) ◽  
pp. 319-336 ◽  
Author(s):  
Luisa Carini ◽  
Vincenzo De Filippis ◽  
Onofrio Mario Di Vincenzo
Keyword(s):  
Commutative Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Generalized Derivations ◽  
Multilinear Polynomials

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, Z(R) the center of R, f(x1,…,xn) a non-central multilinear polynomial over K, d and δ derivations of R, a and b fixed elements of R. Denote by f(R) the set of all evaluations of the polynomial f(x1,…,xn) in R. If a[d(u),u] + [δ (u),u]b = 0 for any u ∈ f(R), we prove that one of the following holds: (i) d = δ = 0; (ii) d = 0 and b = 0; (iii) δ = 0 and a = 0; (iv) a, b ∈ Z(R) and ad + bδ = 0. We also examine some consequences of this result related to generalized derivations and we prove that if d is a derivation of R and g a generalized derivation of R such that g([d(u),u]) = 0 for any u ∈ f(R), then either g = 0 or d = 0.

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