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Пусть $R$ - простое кольцо характеристики, отличной от $2$, $Z$ - центр $R$, $C$ - его расширенный центроид, $L$ - идеал Ли в $R$, $\alpha$ и $\beta$ - два нетривиальных автоморфизма $R$. Предположим, что существуют такие фиксированные целые числа $m,n\ge 1$, что $\alpha(u)^n+\beta(u)^m=0$ для всех $u\in L$. Показано, что в этом случае либо $L$ является центральным, либо $R\subseteq M_2(C)$ ($M_2(C)$ - кольцо $2\times 2$ матриц над $C$), $L$ коммутативен и $u^2\in Z$ для всех $u\in L$. В частности, если $L=[R,R]$, то $R$ коммутативно. Библиография: 22 названия.

Let $R$ be a prime ring, $Q_r$ its right Martindale quotient ring, $L$ a non-central Lie ideal of $R$, $n\geq 1$ a fixed integer, $F$ and $G$ two generalized skew derivations of $R$ with the same associated automorphism, $p\in R$ a fixed element. If $p\bigl(F(x)F(y)-G(y)x\bigr)^n=0$, for any $x,y \in L$, then there exist $a,c\in Q_r$ such that $F(x)=ax$ and $G(x)=cx$, for any $x\in R$, with $pa=pc=0$, unless when $R$ satisfies the standard polynomial identity $s_4(x_1,\ldots,x_4)$.

Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W S . The coannihilator graph of S , denoted by A G ′ S , is an undirected graph with vertex set W S ∗ (set of all nonzero nonunit elements of S ), and α ∼ β is an edge of A G ′ S ⇔ α ∉ α β S or β ∉ α β S , where δ S denotes the principal ideal generated by δ ∈ S . In this study, we first classify finite ring S , for which A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring S , for which A G ′ S is toroidal or projective.

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A . It is denoted by ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

Let R be a prime ring of characteristic different from two, Q be the Martindale quotient ring of R and C be the extended centroid of R. Suppose that d is a derivation of R, I a nonzero ideal of R and \(m,k\in \mathbb {Z^{+}}\) where \(m > 1\). If $$[(x\circ _k y)^d,x\circ _k y]^m=[(x\circ _k y)^d,x\circ _k y]$$ for all \(x, y \in I\), then either \(d = 0\) or R satisfies \(S_{4}\), the standard identity in four variables. We also extend the result to semiprime rings.

Background Milk-borne zoonotic diseases can be acquired by the consumption of nonpasteurized and infected dairy products. Zoonotic infections present a serious public health concern that is responsible for approximately 2.7 billion deaths annually worldwide. However, little is known about the attitudes and knowledge of the farmers regarding milk-borne zoonosis. Objective This study was performed with an aim to assess the knowledge, attitude, and practices (KAP) of farmers regarding milk-borne zoonosis. Methods This cross-sectional KAP study was conducted in District Muzaffarabad, Azad Jammu and Kashmir, from September 1 to October 30, 2019. A pretested structured questionnaire was used to collect information from respondents regarding different aspects of milk-borne zoonosis. All small dairy farms (n=56) with more than 5 animals in District Muzaffarabad were included in this study. Data were collected from respondents (n=100), with an inclusion criterion of having a dairy experience of more than 6 months. Results The findings show that almost 86% of the farmers were unable to name any milk-borne zoonotic disease. About 45.5% of the farmers were unaware of the fact that milk can be a potential source of disease transmission. None of the respondents had any idea about the pasteurization method, and 50% of them had no habit of checking milk quality. However, 81% of the respondents preferred to use boiled milk. Almost 28% of the farmers with high-level education were able to name at least one milk-borne zoonotic disease. The majority of the respondents (99%) did not receive any formal training about zoonotic diseases. Conclusions According to the study, the overall knowledge of farmers regarding milk-borne zoonosis is not adequate. Despite having a positive attitude, the practices of the respondents regarding milk handling were found to be poor. Awareness about important zoonotic diseases and their source of transmission should be created, and a one-health approach to deal with zoonotic infections should be adopted.

In this manuscript, we discuss the behaviour and nature of generalized derivation $${\mathscr {G}}$$ on a (semi-) prime ring $${\mathscr {R}}$$ satisfying certain differential identities over $${\mathscr {I}}$$ , a nonzero ideal of $${\mathscr {R}}$$ . Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.

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Background: The State of Azad Jammu & Kashmir (AJ&K) is polio free since October 2000.The objectives of our study is to review of existing Acute Flaccid Paralysis Surveillance System in Azad Jammu &Kashmir, identify the strong & weak points of the existing system and suggest course of action for efficient performance of the existing system. Methods: This qualitative & quantitative evaluation was conducted at Provincial Disease Surveillance &Response Unit (PDSRU) Muzaffarabad Azad Jammu & Kashmir during March -April 2019. The database of AFP cases during 2018 was reviewed and relevant stakeholder's interviews were conducted consulting guidelines formulated by the Centre for Disease Control & prevention(CDC) in 2001 for Evaluating Public Health Surveillance Systems. Results: In 2018, a total of 265 AFP cases were registered. The mean age was 65 months (range 01 - 180 months). 59 % (n=157) were male children. 58% of cases were under 05 year's age. Standardized case definition and data format with simple information flow was found. System was flexible enough to incorporate measles and neonatal tetanus cases since 2009. Data quality was excellent (100% zero and monthly reports). A close coordination was observed amongst all relevant stakeholders. Sensitivity was 200%. No polio case was identified and therefore, PPV was zero. Majority of cases were reported by public sector (93%).Sufficient financial as well as skilled human resources were available and hence system found stable. Timeliness of reporting found 90%. Conclusion: The performance of AFP surveillance system in AJ&K is up to the mark. However, there is constant threat of reintroduction of polio virus from adjacent area of Punjab & Khyber Pakhtunkhwa provinces. Highly vigilant AFP surveillance system with capacity of rapid response is the solution. Furthermore, it is vital to sustain the AFP Surveillance till the goal of global polio eradication is achieved.

Let R be a prime ring with the extended centroid C and symmetric Martindale quotient ring $$Q_s(R)$$ . In this paper we prove the following result. Let $$F: R \rightarrow R$$ be a generalized derivation associated with a non-zero derivation d on R and let h be an additive map of R such that $$F(x)x=xh(x)$$ for all $$x\in R$$ . Then either R is commutative or $$F(x)=xp$$ and $$h(x)=px$$ where $$p\in Q_{s}(R)$$ .

Let X be a real or complex Banach space, let L ( X ) be the algebra of all bounded linear operators of X and let A ( X ) ⊆ L ( X ) be a standard operator algebra. Suppose there exists a linear mapping T : A ( X ) → L ( X ) satisfying the relation T(An) = T(A)An−1 − AT(An−2)A − An−1T(A) for all A ∈ A ( X ) , where n > 2 is some fixed integer. Then T is of the form: (i)T(A) = 0 for all A ∈ F ( X ) and (ii) T(A) = BA, for all A ∈ A ( X ) and some B ∈ L ( X ) .

Let R be a prime ring of characteristic different from 2, L be a non-central Lie ideal of R and m, n be the fixed positive integers. If R admits a generalized derivation F associated with a deviation d such that $$F(u^2)^m -d(u)^{2n}\in Z(R)$$ for all $$u \in L$$ , then R satisfies $$s_4$$ , the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

Let R be a prime ring, F be a generalized derivation associated with a derivation d of R and m, n be the fixed positive integers. In this paper we study the case when one of the following holds: (i) F(x) ◦m F(y) = (x ◦ y)n, (ii) F(x)◦md(y) = d(x◦y)n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.

Aim: The aim of this study was to determine the functional results of intertrochanteric fracture managed with dynamic hip screw by noticing union time, infection rate, range of hip movement and limb shortening. Methods: A prospective study was performed in thirty femoral intertrochanteric fractures cases. Using a dynamic hip screw, all these cases were treated. Place and Duration: In the Orthopedics Department, Khyber Teaching, Hospital, MTI, Peshawar for one year duration from March 2018 to March 2019. Results: 68.17 ± 12.49 years was the average range. The injuries occurred more often in 17 (57%) men than in 13 (43%) women. It was found that the injury mechanism mostly was a fall in 17 patients (56.7%) and other cause responsible in 13 (43.3%) was road accidents. All patients were hospitalized six hours later and not a single patient received proper management within 6 hours after the injury. The associated injuries occurred in five patients (16.7%). In (60%) of cases right limb was convoluted and in the left limb (40%). In 2 (6.66%) of cases, superficial infection was noted within two weeks. At the end of the six weeks, all patients have a clean wound. None of our patients had a deep infection. 1 patient (3%) only had limb shortening greater than 2 cm, while 29 patients (97%) had limb shortening less than 2 cm. 96.7% of patients achieved full load bearing ability after 24 weeks. At the end of our study, many of our patients (96.7%) received hip flexion / extension at 121 °. ± 8.8o and 9.33 ± 2.5, internal and external rotation 30.5 ± 7.1 and 30.2 ± 6.9and adduction / abduction 31.8o ± 6.5o and 26.8o ± 5.6o. Overall outcome was good. Keywords: Dynamic hip screw, Intertrochanteric femoral fracture.

In this paper we prove that on a 2-torsion free semiprime ring R every Jordan triple (α,β)-derivation (resp. generalized Jordan triple (α,β)-derivation) on Lie ideal L is an (α,β)-derivation on L (resp. generalized (α,β)-derivation on L)

Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.

Let R be a semiprime ring of characteristic different from 2, C its extended centroid, Z(R) its center, F and G non-zero skew derivations of R with associated automorphism \(\alpha \) and m, n positive integers such that $$\begin{aligned}{}[F(x),G(y)]_m=[x,y]^n ~\mathrm{for \, all}~x,y \in R. \end{aligned}$$ Then R is commutative.

Let R be a prime ring of characteristic different from two with center Z(R). In the present paper we study the case when a generalized derivation F associated with a nonzero derivation d of R satisfies $$[F([x, y]_k), [x, y]_k]\in Z(R)$$ for all x, y in some appropriate subset of R, where k is a fixed positive integer. We obtain a description of the structure of R and information on the form of F in terms of the standard identity $$s_4$$ and the multiplication by the specific element from the extended centroid.

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U, C be the extended centroid of \(R,\, F\) and G be two nonzero generalized derivations of R and \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C which is not central valued on R. If $$\begin{aligned} {[}F(u)u, G(v)v]=0 \end{aligned}$$ for all \(u,v\in f(R)\), then there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \([a, b]=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R.

In this paper we prove the following result. Let R be a n!-torsion free semiprime ring and let f :R → R be an additive mapping satisfying the relation f(x)x n + x n f(x) = 0 for all x ∈ R. In this case f = 0.

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x),d(y)] m =[x,y] for all x,y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer–Werner theorem.

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.

In the present paper it is shown that a prime ring R with center Z satisfies \(s_4\), the standard identity in four variables if R admits a non-identity automorphism \(\sigma \) such that \([u^\sigma ,u]^nu^\sigma \in Z\) for all u in some non-central Lie ideal L of R, whenever \(char(R)>n\) or \(char(R)=0\), where n is a fixed positive integer. This result is in the spirit of theorems such as Posner’s second theorem or the Herstein’s theorem on derivations with central values.

Let R be a prime ring, I a nonzero ideal of R, and a ∈ R. Suppose that σ is a nontrivial automorphism of R such that a{(σ(x ∘ y))n − (x ∘ y)m} = 0 or a{(σ([x,y]))n − ([x,y])m} = 0 for all x,y ∈ I, where n and m are fixed positive integers. We prove that if char(R) > n + 1 or char(R) = 0, then either a = 0 or R is commutative.

Let R be a semiprime ring with involution ∗ and let F, D : R → R be additive mappings satisfying the conditions (i) F(x 2) = F(x)x ∗+x ∗ D(x) and D(x 2) = D(x)x ∗+x ∗ D(x); (ii) F(x n + 1) = F(x)(x ∗) n +x ∗ D(x)(x ∗) n − 1+(x ∗)2 D(x)(x ∗) n − 2+⋯+(x ∗) n D(x) for all x ∈ R. Then, F(x y) = F(y)x ∗+y ∗ D(x) and D(x y) = D(y)x ∗+y ∗ D(x) for all x, y ∈ R.

In the present paper it is shown that if R is a semiprime ring of characteristic different from two which admits a derivation d such that \([d(x^m),d(y^n)]=\pm [x^m, y^n]\) for all \(x,y\in R\), where \(m\ge 1, n\ge 1\) are fixed positive integers, then R is commutative. Further using this result it is established that if \(\mathfrak {A}\) is a semiprime Banach algebra and \(\mathscr {H}_1\) and \(\mathscr {H}_2\) are nonvoid open subsets of \(\mathfrak {A}\) which admits a continuous derivation \(d:\mathfrak {A}\rightarrow \mathfrak {A}\) such that \([d(x^m),d(y^n)]\pm [x^m,y^n]=0\) for all \(x\in \mathscr {H}_1\) and \(y\in \mathscr {H}_2\), where m, n are no longer fixed rather they depend on the pair of elements \( x, y \in \mathfrak {A}\), then \(\mathfrak {A}\) is commutative.

Let \(R\) be a prime ring of characteristic different from \(2\) and \(k\ge 1\) fixed positive integer. If \(R\) admits a generalized derivation \(G\) associated with a deviation \(d\), here we study the following cases: (1) \(G([r_1,r_2]_k)\circ ([r_1,r_2]_k)=0\) (2) \(G([r_1,r_2]_k)\circ G([r_3,r_4]_k)=[r_1,r_2]_k\circ [r_3,r_4]_k\) (3) \(G([r_1,r_2])\circ _k G([r_3,r_4])=0\) (4) \(G([r_1,r_2])\circ _k G([r_3,r_4])=[r_1,r_2]\circ _k [r_3,r_4]\). We obtain a description of the structure of \(R\) and information on the form of \(G\) in terms of the commutativity of \(R\) and the multiplication by a specific element from the extended centroid of \(R\).

Let R be a prime ring with center Z, I a nonzero ideal of R, and a non- trivial automorphism of R such that {(x ◦ y) − (x ◦ y)} n ∈ Z for all x;y ∈ I. If either char(R) > n or char (R) = 0, then R satises s4, the standard identity in 4 variables.

Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f(x 1, . . . , x n ) a multilinear polynomial over C, ρ a nonzero right ideal of R and m > 1 a fixed integer such that $\qquad \left ([d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})]\right )^{m}=[d(f(r_{1},\ldots ,r_{n})),f(r_{1},\ldots ,r_{n})] $ for all r 1, . . . , r n ∈ρ. Then either [f(x 1,…,x n ),x n+1]x n+2 is an identity for ρ or d(ρ)ρ = 0.

The main purpose of this paper is to prove the following result: Let n> 1 be a fixed integer, let R be a n!-torsion free semiprime ring, and let f : R → R be an additive mapping satisfying the relation ( f( x), x)n =( ( ... (( f( x), x) ,x ) , . .. ) ,x )= 0 for all x ∈ R. In this case ( f( x), x )= 0 is fulfilled for all x ∈ R. Since any semisimple Banach algebra (for example, C ∗ algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.

In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

Let (R, ∗) be a 2-torsion free ∗-prime ring with involution ∗, and L � 0b e a ∗-Lie ideal of R. An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F( xy ) = F(x)y + xd(y) holds for all x, y ∈ R. In the present paper, we shall show that when L satisfies any of several identities involving F, then L is central.

The aim of this paper is to prove the next result. Let n > 1be an integer and let R be a n!-torsion free semiprime ring. Suppose thatf : R →R is an additive mapping satisfying the relation [f(x),x n ] = 0for all x ∈R. Then f is commuting on R. 1. Introduction and the maintheoremThroughout, R will represent an associative ring with a center Z(R). Letn > 1 be an integer. A ring R is n-torsion free if nx = 0, x ∈ R, impliesx = 0. The Lie product (or a commutator) of elements x,y ∈ R will be denotedby [x,y] (i.e., [x,y] = xy − yx). Recall that a ring R is prime if aRb = {0},a,b ∈ R, implies that either a = 0 or b = 0. Furthermore, a ring R is calledsemiprime if aRa = {0}, a ∈ R, implies a = 0. We will denote by C and Q theextended centroid and the maximal right ring of quotients of a semiprime ringR, respectively. For the explanation of the extended centroid as well as themaximal right ring of quotients of a semiprime ring we refer the reader to [4].As usual, the socle of a ring R will be denoted by soc(R).An additive mapping D : R → R is called a derivation on R if D(xy) =D(x)y + xD(y) holds for all pairs x,y ∈ R. An additive mapping f : R →R is called centralizing on R if [f(x),x] ∈ Z(R) holds for all x ∈ R. In aspecial case, when [f(x),x] = 0 for all x ∈ R, the mapping f is said to becommuting on R. A classical result of Posner [21] (Posner’s second theorem)states that the existence of a nonzero centralizing derivation on a prime ringforces the ring to be commutative. Posner’s second theorem in general cannotbe proved for semiprime rings as shows the following example. Let R

Let R be a prime ring with characteristic different from 2 and U be a Lie ideal of R. In the paper, we initiate the study of generalized Jordan left derivations on Lie ideals of R and prove that every generalized Jordan left derivation on U is a generalized left derivation on U. Further, it is shown that generalized Jordan left biderivation associated with the left biderivation on U is either U ⊆ Z(R) or a right bicentralizer on U.