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1

Generalized derivations and generalized exponential monomials on hypergroups

Fechner Żywilla, Gselmann Eszter, Székelyhidi László

Opuscula Mathematica

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2023

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Vol. 43, no. 4

493--505

EN

In one of our former papers Endomorphisms of the measure algebra of commutative hypergroups we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra.

2

Lie ideals with generalized derivations and derivations of semiprime rings

Sögütcü Emine Koç, Rehman Nadeem ur, Dernek Eda

Fasciculi Mathematici

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2023

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nr 66

107--120

EN

Let R be a 2-torsion free semiprime ring, L a square-closed Lie ideal of R, ø be a derivation of R and α be an automorphism of R. We will demonstrate in this study that ø(L) = (0), and so ø is a zero map on L if any one of the following holds for all r r ∈ L: (i) ø(r)r = 0 (or r ø(r) = 0) (ii) ø(r) r + r (α (r) - r) = 0, (iii) The mapping r → ø (r) + α (r) is commuting on L. Moreover, if any one of the following are satisfied for two generalized derivations (F, ø) and (H, ξ) of R, then ø is a commuting map on L: (iv) F (r) F (s) - H(rs) ∈ Z (R), (v) F (rs) = ± H (rs), (vi) F (rs) = ±H (sr), for all r, s ∈ L.

3

Development of a highly sensitive HPLC method for the simultaneous determination of eight biogenic amines in aquatic products

Fu Qinyi, Zheng Hongwei, Han Xiangning, Cao Limin, Sui Jianxin

Acta Chromatographica

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2021

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Vol. 33, no. 4

378--386

EN

A highly sensitive high-performance liquid chromatography (HPLC) method was developed for the simultaneous determination of eight biogenic amines in aquatic products. The biogenic amines in the sample were extracted with 5% trichloroacetic acid, derived with dansyl chloride (Dns-Cl) and quantified by a UV detector. The results showed that tryptamine (TRY), 2-phenethylamine (PHE), putrescine (PUT), cadaverine (CAD), histamine (HIS), tyramine (TPY), spermidine (SPD), and spermine (SPM) were effectively separated in 18 min in the range of 0.1–50 mg/kg with a good linear coefficient (r2 > 0.999). The detection limits (LODs) of the eight biogenic amines were 0.007–0.021 mg/kg while the limits of quantification (LOQs) were 0.024–0.069 mg/kg with the recoveries basically between 68 and 123%. The determination of eight biogenic amines in five commercial fermented aquatic products indicating that the developed method could be applied for the simultaneous detection of biogenic amines in multiple aquatic products.

4

Remarks on *-(σ, τ)-Lie ideals of *-prime rings with derivation

Sögütcü E. K., Aydin N., Gölbaşi Ö.

Fasciculi Mathematici

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2018

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Nr 60

161--171

EN

Let R be a *- prime ring with characteristic not 2, U a nonzero *- (σ, τ)-Lie ideal of R, d a nonzero derivation of R. Suppose σ, τ be two automorphisms of R such that σd = dσ, τd = dτ and * commutes with σ, τ, d. In the present paper it is shown that if d(U) ⊆ Z or d2 (U) ⊆ Z, then U ⊆ Z.

5

Derivable maps and generalized derivations on nest and standard algebras

Pan Z.

Demonstratio Mathematica

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2016

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Vol. 49, nr 3

331--344

EN

For an algebra A, an A-bimodule M, and m (…) M, define a relation on A by RA(m, 0) = {(a, b) (…) A x A : amb = 0}. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then (…)L(A,B(X)) is a generalized derivation if and only if (..) is derivable on RA(M,0), for some M (…) B(X). We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A = algN on a Hilbert space H such that P is either left-faithful to N or right-faithful to N﬩, if δ (…)L(A, B)H)) is derivable on RA(P, 0) then δ is a generalized derivation.

6

On basic properties of δ-prime and δ-semiprime rings

Kular K.

Czasopismo Techniczne. Nauki Podstawowe

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2016

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Y. 113, iss. 1-NP

107--119

EN

We provide a self-contained discussion of the notions of δ-primeness and δ-semiprimeness for associative rings, possibly without identity. Some of the facts and properties presented in the article seem less known and quite difficult to find in standard reference sources.

PL

Praca jest "samowystarczalnym” omówieniem pojęć δ-pierwszości i δ-półpierwszości, rozważanych w algebrze nieprzemiennej. Wszystkie udowodnione w pracy twierdzenia stosują się i do pierścieni z jedynką, i do pierścieni bez jedynki. Część faktów i własności przedstawionych w pracy wydaje się mało znana i raczej trudna do odszukania w standardowej literaturze.

7

Higher Order Nonlinearity of Niho Functions

Sun G., Wu C.

Fundamenta Informaticae

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2015

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Vol. 137, nr 3

403--412

EN

The r-th order nonlinearity of Boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers. It is also very useful in coding theory, since it is related to the covering radii of Reed-Muller codes. In this paper we investigate the lower bound of the higher-order nonlinearity of Niho Boolean functions f(x) = tr(λxd) over F2n, where... [formula]

8

On (m, n)-derivations of some algebras

Shen Q., Li J., Guo J.

Demonstratio Mathematica

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2014

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Vol. 47, nr 3

672--694

EN

Let A be a unital algebra, δ be a linear mapping from A into itself and m, n be fixed integers. We call δ an (m, n)-derivable mapping at Z, if mδ(AB) + nδ(BA) = mδ(A)B + mAδ(B) + nδ(B)A + nBδ(A) for all A, B (…) A with AB = Z. In this paper, (m, n)-derivable mappings at 0 (resp. IA (…) 0, I) on generalized matrix algebras are characterized. We also study (m, n)-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations.

9

On Jordan triple α-* centralizers of semiprime rings

Ashraf M., Mozumder M.R., Khan A.

Demonstratio Mathematica

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2014

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Vol. 47, nr 1

130--136

EN

Let R be a 2-torsion free semiprime ring equipped with an involution *. An additive mapping T : R → R is called a left (resp. right) Jordan α-* centralizer associated with a function α : R → R if T(x2) = T(x)α(x*) (resp. T(x2) = α(x*)T(x)) holds for all x (…) R. If T is both left and right Jordan α-* centralizer of R, then it is called Jordan α-* centralizer of R. In the present paper it is shown that if α is an automorphism of R, and T : R → R is an additive mapping such that 2T(xyx) = T(x)α(y*x*) + α(x*y*)T(x) holds for all x; y (…) R, then T is a Jordan α-* centralizer of R.

10

On derivations of operator algebras with involution

Širovnik N., Vukman J.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

784--790

EN

The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) (…) L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A (…) A(X). In this case, D is of the form D(A) = [A,B] for all A (…) A(X) and some fixed B (…) L(X), which means that D is a derivation.

11

Identities with generalized derivations in semiprime rings

Dhara B., Ali S., Pattanayak A.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

453--460

EN

Let R be a semiprime ring. An additive mapping F:R  R is called a generalized derivation of R if there exists a derivation d : R  R such that F(xy) = F(x)y + xd(y) holds, for all x,y  R. The objective of the present paper is to study the following situations: (1) (...), for all x, y in some appropriate subset of R.

12

A characterization of one-element p-bases of rings of constants

Jędrzejewicz P.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 1

19-26

EN

Let Κ be a unique factorization domain of characteristic p > 0, and let ƒ ∈ Κ[χ1,..., χn] be a polynomial not lying in Κ[wzór]. We prove that Κ[wzór] is the ring of constants of a Κ-derivation of Κ[χ1,..., χn] if and only if all the partial derivatives of ƒ are relatively prime. The proof is based on a generalization of Freudenburg's lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

13

Derivation with engel conditions on multilinear polynomials in prime rings

Dhara B.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

467-478

EN

Let R be a prime ring with extended centroid C and characteristic different from 2, d a nonzero derivation of R, f(x1,..., xn) a nonzero multilinear polynomial over C such that [d2(f(x1,... , xn)),d(f(x1,... ,xn))]k = 0 for all x1,... ,xn in some nonzero right ideal [...] of R, where k is a fixed positive integer. If d(p) p [..]0, then pC = eRC for some idempotent e in the socle of RC and /(x1,..., xn) is central-valued on eRCe.

14

Identities with two automorphisms on semiprime rings

Kosi-Ulbl I.

Demonstratio Mathematica

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2007

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Vol. 40, nr 1

37-42

EN

In this paper we investigate identities with two automorphisms on semiprime rings. We prove the following result: Let T, S : R approaches R be automorphisms where R is a 2-torsion free semiprime ring satisfying the relation T(x)x = xS(x) for all x is an element of R. In this case the mapping x approaches T(x) - x maps R into its center and T = S.

15

On some equations related to derivations in rings and Banach algebras

Vukman J., Kosi-Ulbl I.

Demonstratio Mathematica

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2006

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Vol. 39, nr 1

61-66

EN

The main purpose of this paper is to investigate additive mapping D : R -> R, where R is a (m + n +1)! and \m2 + n2 - m - n - 4mn\ -torsion free semiprime ring with the identity element, satisfying the relation 2D(xm+n+l) = (m+-n+1)(xmD(x)xn +-xnD(x)xm), for all is an element of R and some integers m > 1, n > 1, m2 + n2 - m - n - 4mn /=0.

16

Identities with products of (alpha, beta)-derivations on prime rings

Vukman J.

Demonstratio Mathematica

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2006

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Vol. 39, nr 2

291-298

EN

The main purpose of this paper is to prove the following result. Let R be a noncommutative prime ring of characteristic different from two and let D and G = 0 be (\alpha, beta)-derivations of R into itself such that G commutes with alpha and beta. If [D{x), G(x)] = 0 holds for all x is an eleemnt of R then D = lambdaG where lambda is an element from the extended centroid of R.

17

On alfa-derivations of prime and semiprime rings

Vukman J.

Demonstratio Mathematica

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2005

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Vol. 38, nr 2

283--290

EN

In this paper we investigate identities with alfa-derivations on prime and semiprime rings. We prove, for example, the following result. If D : R - R is an alfa-derivation of a 2 and 3-torsion free semiprime ring R such that [D(x},x2] = 0 holds, for all x is an element of R, then D maps R into its center. The results of this paper are motivated by the work of Thaheem and Samman [20].

18

Regular derivation and synonymy in an e-dictionary of Serbian

Vitas D., Krstev C.

Archives of Control Sciences

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2005

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Vol. 15, no. 3

469-480

EN

In this paper we explore the relation between derivational morphology and synonymy in connection with an electronic dictionary, inspired by the work of Maurice Gross. The characteristics of this relation are illustrated by derivation in Serbian, which produces new lemmas with predictable meaning. We call this regular derivation. We then demonstrate how this kind of derivation is handled in text processing using a morphological e-dictionary of Serbian and a collection of transducers with lexical constraints. Finally, we analyze the cases of synonymy that include regular derivation in one aligned text.

19

On (α, β)-derivations of semiprime rings, II

Chaudhry M., Thaheem A.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

793--802

20

On (α, β)-derivations of semiprime rings

Chaudhry M. A., Thaheem A. B.

Demonstratio Mathematica

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2003

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Vol. 36, nr 2

283--287

EN

We show that if α and β are centralizing automorphisms and d a centralizing (α, β)-derivation of a semiprime ring R, then d is commuting. Some results on α-derivations and centralizing derivations of semiprime rings follow as applications of this result.