Volume 23, Issue 1

Let L = ( L ,∧,∨) be a lattice with a least element called zero and denoted by 0. The annihilating-ideal graph of L , denoted by 𝔸𝔾( L ), is a graph whose vertex-set is the set of all non-trivial ideals of L and, for every two distinct vertices I and J , I is adjacent to J if and only if I ∧ J = {0}. In this paper, we study some properties of 𝔸𝔾( L ). Also, we completely determine when the annihilating-ideal graph is complete bipartite, split and end-regular.

In [Acta Math. Hungar. 66 (1995), 337–343], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d(xy)=d(yx)${d(xy)=d(yx)}$ for all x,y∈R${x,y\in R}$, then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d(xx*)=d(x*x)${d(xx^*)=d(x^*x)}$ for all x ∈ R and S(R)∩Z(R)≠(0)${S(R)\cap Z(R)\ne (0)}$, then R is commutative. Moreover, some related results have also been discussed.

Let R be a semiprime ring with center Z and extended centroid C . For a fixed integer n ≥ 2, the trace δ:R→R${\delta \colon R\rightarrow R}$ of a permuting n -additive mapping D:Rn→R${D\colon R^n\rightarrow R}$ is defined as δ(x)=D(x,...,x)${\delta (x)=D(x,\ldots ,x)}$ for all x ∈ R . The notion of permuting n -derivation was introduced by Park [J. Chungcheong Math. Soc. 22 (2009), no.3, 451–458] as follows: a permuting n -additive mapping Δ:Rn→R${\Delta \colon R^n\rightarrow R}$ is said to be permuting n -derivation if Δ(x1,x2,⋯,xixi',⋯,xn)=Δ(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.$ \Delta (x_1,x_2,\dots , x_ix_i^{\prime },\dots , x_n)=\Delta (x_1,x_2,\dots , x_i,\dots , x_n)x_i^{\prime }+ x_i\Delta (x_1,x_2,\dots , x_i^{\prime },\dots , x_n)\quad \text{for all }x_i ,x_i^{\prime } \in R. $ A permuting n -additive mapping Ω:Rn→R${\Omega \colon R^n\rightarrow R}$ is known to be a permuting generalized n -derivation if there exists a permuting n -derivation Δ:Rn→R${\Delta \colon R^n\rightarrow R}$ such that Ω(x1,x2,⋯,xixi',⋯,xn)=Ω(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.$ \Omega (x_1,x_2,\dots , x_ix_i^{\prime },\dots , x_n)=\Omega (x_1,x_2,\dots , x_i,\dots , x_n)x_i^{\prime }+ x_i\Delta (x_1,x_2,\dots , x_i^{\prime },\dots , x_n)\quad \text{for all }x_i ,x_i^{\prime } \in R. $ The main result of this paper states that if I is a nonzero ideal of a semiprime ring R and Δ:Rn→R${\Delta :R^n\rightarrow R}$ is a permuting n -derivation such that Δ(I,...,I)≠{0}${\Delta (I,\ldots ,I)\ne \lbrace 0\rbrace }$ and [δ(x),x]=0${[\delta (x),x]=0}$ for all x ∈ I , where δ is the trace of Δ, then R contains a nonzero central ideal. Furthermore, some related results are also proven.

The paper deals with the measurability properties of some classical subsets of the real line ℝ having an extra-ordinary descriptive structure: Vitali sets, Bernstein sets, Hamel bases, Luzin sets and Sierpiński sets. In particular, it is shown that there exists a translation invariant measure μ on ℝ extending the Lebesgue measure and such that all Sierpiński sets are measurable with respect to μ.

Let (umns)${(u_{mns})}$ be a ( C ,1,1,1) summable triple sequence of real numbers. We give one-sided Tauberian conditions of Landau and Hardy type under which (umns)${(u_{mns})}$ converges in Pringsheim's sense. We prove that (umns)${(u_{mns})}$ converges in Pringsheim's sense if (umns)${(u_{mns})}$ is slowly oscillating in certain senses. Moreover, we extend a Tauberian theorem given by Móricz [Studia Math. 110 (1994), 83–96] for double sequences to triple sequences.

In this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

We give general results which provide asymptotic irrationality measures and estimations for the denominators of the convergents for certain almost periodic simple continued fraction expansions. As an application we obtain new irrationality measures, for example, to Napier's constant e , certain powers of e , tanh1${\tanh 1}$ and a ratio of modified Bessel functions.

The paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

The purpose of this paper is to study the relationships between an n -Hom-Lie algebra and its induced (n+1)-Hom-Lie algebra. We provide an overview of the theory and explore structure properties such as ideals, centers, derived series, solvability, nilpotency, central extensions, and the cohomology.

In this paper, we establish a new integral identity for n -times differentiable functions defined on an invex subset of ℝ. Hermite–Hadamard type integral inequalities for n -times differentiable preinvex functions are then established by using this identity and Hölder's inequality.

In this paper, we introduce new nonlinear iterative algorithms. These algorithms are used to study the convergence of generated iterative sequences in modular spaces. Moreover, we introduce a new double sequence iteration and prove that sequences converge strongly to a fixed point of a ρ-quasi contraction mapping in modular spaces. Finally, some illustrative numerical examples (using the Matlab software) are presented.

In this paper, we derive general solution of new n -dimensional quintic and sextic functional equations and investigate the Hyers–Ulam stability, Hyers–Ulam–Rassias stability and generalized Hyers–Ulam–Rassias stability for quintic and sextic functional equations in Felbin type fuzzy normed linear spaces. Also, we give the counter examples for the Hyers–Ulam–Rassias stability of quintic and sextic functional equations for some cases.

We give some new characterizations of quasi-Frobenius rings by means of YJ-injective rings and JGP-injective rings, respectively. For example, we show that a two-sided YJ-injective right noetherian ring is quasi-Frobenius, which gives an affirmative answer to an open question asked by Roger Yue Chi Ming; a right CF, semiregular, right JGP-injective ring is quasi-Frobenius, which improves the result on p -injective rings obtained by Jianlong Chen. Moreover, a series of conditions is given under which a left noetherian left mininjective right JGP-injective ring is quasi-Frobenius.