TOWERS OF DERIVATION FOR LIE RINGS AND SOME RESULTS ON COMPLETE LIE RINGS

The well-known tower theorem of groups （resp. Lie algebras） shows that the tower of automorphism groups （resp. derivation algebras） of a finite group （resp. a finite dimensional Lie algebra） with trivial center terminates after finitely many steps. We generalize these results for Lie rings, and present some necessary and sufficient conditions for Lie rings to be complete.

Cocentralizing Generalized Derivations in Prime Rings

Let R be a prime ring with center Z and S包含 R. Two mappings D and G of R into itself are called cocentralizing on S if D（x）x - xG（x） ∈ Z for all x ∈S. The main purpose of this paper is to describe the structure of generalized derivations which are cocentralizing on ideals, left ideals and Lie ideals of a prime ring, respectively. The semiprime ease is also considered.

The Linear Combination of Derivations in Prime Rings

In this paper we discuss the linear combination of derivations in prime rings which was initiated by Niu Fengwen. Our aim is to improve Niu's result. For the proof of the main theorem we generalize a result of Bresar and obtain a lemma that is of independent interest.

Generalized Derivations in Prime Rings

Let R be a ring, a ,b ∈ R, ( D , α ) and (G , β ) be two generalized derivations of R . It is proved that if aD ( x ) = G ( x )b for all x ∈ R, then one of the following possibilities holds: (i) If either a or b is contained in C , then α = β= 0 and there exist p , q ∈ Qr ( RC) such that D ( x )= px and G ( x )= qx for all x ∈ R;(ii) If both a and b are contained in C , then either a = b= 0 or D and G are C-linearly dependent;(iii) If neither a nor b is contained in C , then there exist p , q ∈ Qr ( RC) and w ∈ Qr ( R) such that α ( x ) = [ q ,x] and β ( x ) = [ x ,p] for all x ∈ R, whence D ( x )= wx-xq and G ( x )= xp + avx with v ∈ C and aw-pb= 0.

Compositions of Generalized Derivations in Prime Rings

Let R be a prime ring with a non-central Lie ideal L. In the present paper we show that if the composition DG of two generalized derivations D and G is zero on L, then DG must be zero on R. And we get all the possibilities for the composition of a couple of generalized derivations to be zero on a non-central Lie ideal of a prime ring.

Left Annihilator of Power Values with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring with characteristic di erent from two,d a derivation of R,L a noncentral Lie ideal of R,and a2R.In the present paper it is showed that if a(d(u^m)±u^m)^n for all u∈L,where m;n are xed positive integers,then a=0 unless R satis es s4,the standard polynomial identity in four variables.

Annihilator on Co-commutators with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring of characteristic different from 2, d and 9 two derivations of R at least one of which is nonzero, L a non-central Lie ideal of R, and a ∈ R. We prove that if a（d（u）u - ug（u）） = 0 for any u ∈ L, then either a = O, or R is an sa-ring, d（x） = [p, x], and g（x） = -d（x） for some p in the Martindale quotient ring of R.

Derivations and 2-Local Derivations on the BMS-Weyl Algebra

Using the theory of derivations on finitely generated and graded Lie algebras, we determine that derivations of the BMS-Weyl algebra are all inner. On this basis, it is proved that every 2-local derivation of the BMS-Weyl algebra is a derivation.

Weak M-Armendariz rings

For a monoid M, this paper introduces the weak M- Armendariz rings which are a common generalization of the M- Armendariz rings and the weak Armendariz rings, and investigates their properties. Moreover, this paper proves that： a ring R is weak M-Armendariz if and only if for any n, the n-by-n upper triangular matrix ring Tn （R） over R is weak M- Armendariz; if I is a semicommutative ideal of ring R such that R/I is weak M-Armendariz, then R is weak M-Armendariz, where M is a strictly totally ordered monoid; if a ring R is semicommutative and M-Armendariz, then R is weak M × N- Armendariz, where N is a strictly totally ordered monoid; a finitely generated Abelian group G is torsion-free if and only if there exists a ring R such that R is weak G-Armendariz.

Rising utilization of stable isotopes in tree rings for climate change and forest ecology

Analyses of stable isotopes(C,O,H)in tree rings are increasingly important cross-disciplinary programs.The rapid development in this field documented in an increasing number of publications requires a comprehensive review.This study includes a bibliometric analysis-based review to better understand research trends in tree ring stable isotope research.Overall,1475 publications were selected from the Web of Science Core Collection for 1974-2023.The findings are that:(1)numbers of annual publications and citations increased since 1974.From 1974 to 1980,there were around two relevant publications per year.However,from 2020 to 2022,this rose sharply to 109 publications per year.Likewise,average article citations were less than four per year before 1990,but were around four per article per year after 2000;(2)the major subjects using tree ring stable isotopes include forestry,geosciences,and environmental sciences,contributing to 42.5%of the total during 1974-2023;(3)the top three most productive institutions are the Chinese Academy of Sciences(423),the Swiss Federal Institute for Forest,Snow and Landscape Research(227),and the University of Arizona(204).These achievements result from strong collaborations;(4)review papers,for example,(Dawson et al.,Annu Rev Ecol Syst 33:507-559,2002)and(McCarroll and Loader,Quat Sci Rev 23:771-801,2004),are among the most cited,with more than 1000 citations;(5)tree ring stable isotope studies mainly focus on climatology and ecology,with atmospheric CO_(2) one of the most popular topics.Since 2010,precipitation and drought have received increasing attention.Based on this analysis,the research stages,key findings,debated issues,limitations and direc-tions for future research are summarized.This study serves as an important attempt to understand the progress on the use of stable isotopes in tree rings,providing scientific guid-ance for young researchers in this field.

Cover story: Neuromodulation Brings Sunshine to Diseased Brains

Various invasive and non-invasive brain modulation methods are widely used to regulate sensory,cognitive,and motor functions both within local and large brain networks.This special issue provides innovative studies to optimize relevant methods and understand their neural mechanisms,paving the way for personalized neuromodulation of cognitive function.The cover of this issue uses the image of the four heavenly Kings in Chinese culture who are in charge of good weather.It compares the process of neuroregulation technology that integrates the principles of sound,light electricity or magnetism in treating the diseased brain,to the moment when these gods cast their spells to bring sunshine and peace to the human brain in the midst of lightning and thunder.

Characterizations of semihoops based on derivations

In this paper,we discuss the related properties of some particular derivations in semihoops and give some characterizations of them.Then,we prove that every Heyting algebra is isomorphic to the algebra of all multiplicative derivations and show that every Boolean algebra is isomorphic to the algebra of all implicative derivations.Finally,we show that the sets of multiplicative and implicative derivations on bounded regular idempotent semihoops are in oneto-one correspondence.

Derivations of Certain Lie Algebras of Upper Triangular Matrices over Commutative Rings

Let R be an arbitrary commutative ring with identity. Denote by t the Lie algebra over R consisting of all upper triangular n by n matrices and let b be the Lie subalgebra of t consisting of all matrices of trace 0. The aim of this paper is to give an explicit description of the derivation algebras of the Lie algebras t and b, respectively.

Commutativity of Near-rings with Derivations

In this paper we first prove that a near-ring admits a derivation if and only if it is zero-symmetric. Also, we prove some commutativity theorems for a non-necessarily 3-prime near-ring R with a suitably-constrained derivation d satisfying the condition that d（a） is not a left zero-divisor in R for some a ∈ R. As consequences, we generalize several commutativity theorems for 3-prime near-rings admitting derivations.

GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS

Let P be a parabolic subalgebra of a general linear Lie algebra gl（n,F） over a field F, where n ≥ 3, F contains at least n different elements, and char（F） ≠ 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.

On the Derivations of Semiprime Rings and Noncommutative Banach Algebras

Let A be a noncommutative Banach algebra.Suppose there exists a continuous linear Jordan derivation D:A→A such that [D(x),x]D(x)[D(x),x]∈ rad(A) for all x ∈ A.In this case, D(A)rad(A).

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K： A →R be additive maps such that F（[x, y]） = F（x）y - yg（x） - T（y）x ＋ xD（y） for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg（R） 〉 3 and also in the case A is a noncentral Lie ideal and deg（R） 〉 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.

Jordan Left Derivations of Generalized Matrix Algebras

In this paper, it is proved that under certain conditions, each Jordan left derivation on a generalized matrix algebra is zero and each generalized Jordan left derivation is a generalized left derivation.

Derivations on Semiprime Rings

Let R be a semiprime ring with the center Z(R), d and g be derivations of R, L be a nonzero left ideal of R and rR(L) = 0. Suppose that d(x)x - xg(x) ∈ Z(R) for all x ∈ L, then d(R) Z(R) and the ideal of R generated by d(R) is in the center of R.

Effectivity and safety of trifocal intraocular lenses and capsular tension rings implantation for cataract patients with axial high myopia

●AIM:To assess effectivity and safety of trifocal intraocular lenses(IOLs)and capsular tension rings in treating cataract patients with axial high myopia.●METHODS:A prospective nonrandomized controlled clinical trial was conducted.Totally 98 eyes(74 patients)who underwent femtosecond laser-assisted cataract surgery(FLACS)with trifocal IOLs were enrolled in the study and followed up for 2y after surgery:46 eyes(33 patients)with capsular tension ring implantation in the long axial lengths(AL)group(26<AL<29 mm)and 52 eyes(41 patients)in the normal AL group(22<AL<24.5 mm).Postoperative outcomes about effectivity and safety,including the subjective and objective visual quality,and postoperative complications were assessed.●RESULTS:Uncorrected distance visual acuity at 5 m and uncorrected intermediate visual acuity at 60 and 80 cm in the long AL group were significantly worse than those in the normal AL group at 3mo postoperatively(P<0.05).The differences in reading speed,spectacle independence and potential visual complaints between the two groups were not statistically significant(P>0.05).The dysfunctional lens index and total modulation transfer function(MTF)average height were similar between the two groups.The postoperative internal coma aberrations in the axial high myopia eyes were significantly higher than that in the normal AL group(P<0.05).The total satisfaction score in the long AL group(91.32±2.76)was slightly higher than that in the normal AL group(90.36±3.47),but there was no difference(P=0.136).A statistically negative correlation was found between corrected distance visual acuity(CDVA)and dysfunctional lens index(r=-0.382,P=0.009),and between CDVA and the total MTF average height(r=-0.374,P=0.01).But there was no significant correlation between CDVA and total satisfaction score(r=0.059,P=0.696).Postoperative complications mainly presented as posterior capsular opacity(PCO),retinal detachment and cystoid macular edema.There was no difference in the incidence of fundus disease(6.5%vs 3.8%,P=0.663)or PCO(17.4%vs 7.7%,P=0.217)between the two groups at two years.●CONCLUSION:The utilization of trifocal IOL and capsular tension ring implantation is beneficial for cataract patients with axial high myopia undergoing FLACS.This approach not only ensures excellent subjective feelings and objective visual quality,but also does not increase the incidence of postoperative complications.