# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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• ### Published online May 9, 2022

#### Rings with a right duo factor ring by an ideal contained in the center

Jeoung Soo Cheon, Tai Keun Kwak, Yang Lee, Zhelin Piao, and Sang Jo Yun

Abstract : This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We fi rst see that this property is seated between right duo and right pi-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D3(R) is right CIFD, where D3(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.

• ### Published online May 11, 2022

#### The characterisation of BMO via commutators in variable Lebesgue spaces on stratified groups

Dongli Liu, Jian Tan, and Jiman Zhao

Abstract : Let T be a bilinear Calder\'{o}n-Zygmund operator, b\in \cup_{q>1}L_{loc}^{q}(G). We firstly obtain a constructive proof of the weak factorisation of Hardy spaces, then we establish the characterization of BMO spaces by the boundedness of the commutator [b, T]_{j} in variable Lebesgue spaces.

• ### Published online May 11, 2022

#### Strong Classification of Extensions of Classifiable $C^*$-algebras

Søren Eilers, Gunnar Restorff, and Efren Ruiz

Abstract : We show that certain extensions of classifiable $C^*$-algebras are strongly classified by the associated six-term exact sequence in $K$-theory together with the positive cone of $K_0$-groups of the ideal and quotient. We use our results to completely classify all unital graph $C^*$-algebras with exactly one non-trivial ideal.

• ### Published online May 12, 2022

#### On the sizes of dual groups

Joungmin Song

Abstract : We give a formula for the sizes of the dual groups. It is obtained by generalizing a size estimation of certain algebraic structure that lies in the heart of the proof of the celebrated primality test by Agrawal, Kayal and Saxena. In turn, by using our formula, we are able to give a streamlined survey of the AKS test.

• ### Published online May 16, 2022

#### Constructions of regular sparse anti-magic squares

Guangzhou Chen, Wen Li, Bangying Xin, and Ming Zhong

Abstract : For positive integers $n$ and $d$ with $d Show More • ### Published online May 11, 2022 #### Characterizing S-flat modules and S-von Neumann regular rings by uniformity Xiaolei Zhang Abstract : Let$R$be a ring and$S$a multiplicative subset of$R$. An$R$-module$T$is called uniformly$S$-torsion provided that$sT=0$for some$s\in S$. The notion of$S$-exact sequences is also introduced from the viewpoint of uniformity. An$R$-module$F$is called$S$-flat provided that the induced sequence$0\rightarrow A\otimes_RF\rightarrow B\otimes_RF\rightarrow C\otimes_RF\rightarrow 0$is$S$-exact for any$S$-exact sequence$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$. A ring$R$is called$S$-von Neumann regular provided there exists an element$s\in S$satisfies that for any$a\in R$there exists$r\in R$such that$sa=ra^2$. We obtain that a ring$R$an$S$-von Neumann regular ring if and only if any$R$-module is$S$-flat. Several properties of$S$-flat modules and$S$-von Neumann regular rings are obtained. Show More • ### Published online March 8, 2022 #### Free products of operator systems Florin Pop Abstract : In this paper we introduce the notion of universal free product for operator systems and operator spaces, and prove extension results for the Operator System Lifting Property (OSLP) and Operator System Local Lifting Property (OSLLP) to the universal free product. • ### Published online March 8, 2022 #### On toric Hamiltonian T-spaces with anti-symplectic involutions Jinhong Kim Abstract : The aim of this paper is to deal with the realization problem of a given Lagrangian submanifold of a symplectic manifold as the fixed point set of an anti-symplectic involution. To be more precise, let$(X, \omega, \mu)$be a toric Hamiltonian$T$-space, and let$\Delta=\mu(X)$denote the moment polytope. Let$\tau$be an anti-symplectic involution of$X$of$X$such that$\tau$maps the fibers of$\mu$to (possibly different) fibers of$\mu$, and let$p_0$be a point in the interior of$\Delta$. If the toric fiber$\mu^{-1}(p_0)$is real Lagrangian with respect to$\tau$, then we show that$p_0$should be the origin and, furthermore,$\Delta$should be centrally symmetric. In this paper, we also provide a simple example asserting that the condition of$\tau$preserving the fibration structure of$\mu$plays a crucial role in the proof of our main result, which thus disproves a general question stated without any restriction about the fibration structure of$\mu$. Show More • ### Published online May 11, 2022 #### On Eigensharpness and almost Eigensharpness of Lexicographic Products of Some Graphs Ahmad Abbasi and Mona Gholamnia Taleshani Abstract : The minimum number of complete bipartite subgraphs needed to partition the edges of a graph G is denoted by b(G). A known lower bound on b(G) states that b(G) > max {p(G), q(G)}, where p(G) and q(G) are the numbers of positive and negative eigenvalues of the adjacency matrix of G, respectively. When equality is attained, G is said to be eigensharp and when b(G) = max {p(G), q(G)} + 1, G is called an almost eigensharp graph. In this paper, we investigate the eigensharpness and almost eigensharpness of lexicographic product of some graphs. • ### Published online May 12, 2022 #### Existence of the continued fractions of$\sqrt{d}$and its applications Jun Ho Lee Abstract : It is well known that the continued fraction expansion of$\sqrt{d}$has the form$[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]$and$a_1, \ldots, a_{l-1}$is a palindromic sequence of positive integers. For a given positive integer$l$and a palindromic sequence of positive integers$a_1, \ldots, a_{l-1}$, we define the set$S(l;a_1, \ldots, a_{l-1}) :=\{d\in \mathbb{Z} \,| \, d>0, \sqrt{d}=[a_0, \overline{a_1, \ldots, a_{l-1}, 2a_0}]\}$. In this paper, we completely determine when$S(l;a_1, \ldots, a_{l-1})$is not empty in the case that$l$is$4$,$5$,$6$, or$7$. We also give similar results for$(1+\sqrt{d})/2$. For the case that$l$is$4$,$5$, or$6$, we explicitly describe the fundamental units of the real quadratic field$\mathbb{Q}(\sqrt{d})$. Finally, we apply our results to the Mordell conjecture for the fundamental units of$\mathbb{Q}(\sqrt{d})$. Show More • ### Published online March 15, 2022 #### Finiteness and vanishing results on hypersurfaces with finite index in$\mathbb{R}^{n+1}$: a revision Nguyen Van Duc Abstract : In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in$\mathbb{R}^{n+1}$. When the hypersurface is stable minimal, we show that there is no nontrivial$L^{2p}$harmonic$1$-form for some$p$. The our range of$p$is better than those in \cite{DS}. With the same range of$p$, we also give finiteness results on minimal hypersurfaces with finite index. • ### Published online March 10, 2022 #### A generalization of w-linked extensions Xiaoying Wu Abstract : In this paper, the concepts of w-linked homomorphisms, the wφ-operation, and DWφrings are introduced. Also the relationships between wφ-ideals and w-ideals over a w-linked homomorphism φ : R → T are discussed.More precisely, it is shown that every wφ-ideal of T is a w-ideal of T. Besides,it is shown that if T is not a DWφring, then T must have an infinite number of maximal wφ-ideals. Finally we give an application of Cohen’s Theorem over w-factor rings, namely it is shown that an integral domain R is an SM-domain with w-dim(R) ≤ 1, if and only if for any nonzero w-ideal I of R, (R/I)w is an Artinian ring, if and only if for any nonzero element a ∈ R, (R/(a))w is an Artinian ring, if and only if for any nonzero element a ∈ R, R satisfies the descending chain condition on w-ideals of R containing a. Show More • ### Published online May 12, 2022 #### Application of Rothe's method to a nonlinear wave equation on graphs Yong Lin and Yuanyuan Xie Abstract : We study a nonlinear wave equation on finite connected weighted graphs. Using Rothe's and energy methods, we prove the existence and uniqueness of solution under certain assumption. For linear wave equation on graphs, Lin and Xie \cite{Lin-Xie} obtained the existence and uniqueness of solution. The main novelty of this paper is that the wave equation we considered has the nonlinear damping term$|u_t|^{p-1}\cdot u_t$($p>1$). • ### Published online March 10, 2022 #### Weighted integral inequalities for modified integral Hardy operators Duranta Chutia and Rajib Haloi Abstract : In this article, we study the weak and extra-weak type integral inequalities for the modified integral Hardy operators. We provide suitable conditions on the weights$\omega, \rho, \phi$and$\psito hold the following weak type modular inequalities. \begin{align*} \mathcal{U}^{-1} \bigg ( \int \limits_{ \{ | \mathcal{I}f | > \gamma\}} \mathcal{U} \Big(\gamma \omega \Big ) \rho \bigg ) & \leq \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \Big ( C |f| \phi\Big) \psi \bigg ), \end{align*} where\mathcal{I}is the modified integral Hardy operators . We also obtain a necesary and sufficient condition for the following extra-weak type integral inequalities. \begin{align*} \omega \bigg ( \Big\{ |\mathcal{I}f| > \gamma \Big \} \bigg) &\leq \mathcal{U}\circ \mathcal{V}^{-1} \bigg ( \int_{0}^{\infty} \mathcal{V} \bigg ( \dfrac{C |f| \phi}{\gamma} \bigg) \psi \bigg ). \end{align*} Further, we discuss the above two inequalities for the conjugate of the modified integral Hardy operators. It will extend the existing results for the Hardy operators and its integral version. Show More • ### Published online March 10, 2022 #### Computation of Wedderburn decomposition of groups algebras from their subalgebra Gaurav Mittal and Rajendra Sharma Abstract : In this paper, we show that under certain conditions the Wedderburn decomposition of a finite semisimple group algebra\mathbb{F}_qG$can be deduced from a subalgebra$\mathbb{F}_q(G/H)$of factor group$G/H$of$G$, where$H$is a normal subgroup of$G$of prime order$P$. Here, we assume that$q=p^r$for some prime$p$and the center of each Wedderburn component of$\mathbb{F}_qG$is the coefficient field$\mathbb{F}_q$. • ### Published online May 12, 2022 #### A sharp integral inequality for compact linear Weingarten hypersurfaces Henrique de Lima, Fábio dos Santos, and Lucas Rocha Abstract : We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces immersed in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvature, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in reference [3] for the case of hypersurfaces with constant scalar curvature in the Euclidean round sphere. • ### Published online March 10, 2022 #### Uniqueness of meromorphic solutions of a certain type of difference equations Jun-Fan Chen and Shu-Qing Lin Abstract : In this paper, we study the uniqueness of two finite order transcendental meromorphic solutions$f(z)$and$g(z)$of the following complex difference equation $$A_{1}(z)f(z+1)+A_{0}(z)f(z)=F(z)e^{\alpha(z)}$$ when they share 0,$\infty$CM, where$A_{1}(z),A_{0}(z),F(z)$are non-zero polynomials,$\alpha(z)$is a polynomial. Our result generalizes and complements some known results given recently by Cui and Chen, Li and Chen. Examples for the precision of our result are also supplied. • ### Published online March 7, 2022 #### BEZOUT RINGS AND WEAKLY BEZOUT RINGS Haitham EL ALAOUI Abstract : In this paper, we study some properties of Bézout and weakly Bézout rings. Then, we investigate the transfer of these notions to trivial ring extensions and amalgamated algebras along an ideal. Also, in the context of domains we show that the amalgamated is an Bézout ring if and only if its a weakly Bézout ring. All along the paper, we put the new results to enrich the current literature with new families of examples of non-Bézout weakly Bézout rings. • ### Published online May 12, 2022 #### On right regularity of commutators Da Woon Jung, Chang Ik Lee, Yang Lee, Sangwon Park, Sung Ju Ryu, and Hyo Jin Sung Abstract : We study the structure of right regular commutators, and call a ring$R${\it strongly C-regular} if$ab-ba\in (ab-ba)^2R$for any$a, b\in R$. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring$R$, (i) if$R/W(R)$is commutative then$R$is commutative; and (ii) every prime factor ring of$R$is either a commutative domain or a noncommutative division ring, where$W(R)$is the Wedderburn radical of$R$. Show More • ### Published online March 16, 2022 #### Knots in homology lens spaces determined by their complements Kazuhiro Ichihara and Toshio Saito Abstract : In this paper, we consider the knot complement problem for not null-homologous knots in homology lens spaces. Let$M$be a homology lens space with$H_1(M; \mathbb{Z}) \cong \mathbb{Z}_p$and$K$a not null-homologous knot in$M$. We show that,$K$is determined by its complement if$M$is non-hyperbolic,$K$is hyperbolic, and$p$is a prime more than 7, or, if$M$is actually a lens space$L(p,q)$and$K$represents a generator of$H_1(L(p,q))$. • ### Published online March 16, 2022 #### ON COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM VECTORS IN HILBERT SPACES Vu Thi Ngoc Anh and Nguyen Thi Thanh Hien Abstract : This paper establishes the {Baum--Katz} type theorem and the {Marcinkiewicz--Zymund} type strong law of large numbers for sequences of coordinatewise negatively associated and identically distributed random vectors$\{X,X_n,n\ge1\}$taking values in a Hilbert space$H$with general normalizing constants$b_n=n^{\alpha}\widetilde L(n^{\alpha})$, where$\widetilde L(\cdot)$is the de Bruijn conjugate of a slowly varying function$L(\cdot).$The main result extends and unifies many results in the literature. The sharpness of the result is illustrated by two examples. • ### Published online March 16, 2022 #### The Kähler Different of a Set of Points in$\mathbb{P}^m\times\mathbb{P}^n$Nguyen T. Hoa, Tran N.K. Linh, Le N. Long, Phan T.T. Nhan, and Nguyen T.P. Nhi Abstract : Given an ACM set$\mathbb{X}$of points in a multiprojective space$\mathbb{P}^m\times\mathbb{P}^n$over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for$\mathbb{X}$. In$\mathbb{P}^1\times \mathbb{P}^1$, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kaehler different. However, this result fails to hold in$\mathbb{P}^m\times\mathbb{P}^n$for$n>1$or$m>1$. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that$\mathbb{X}$is a complete intersection of type$(d_1,...,d_m,d'_1,...,d'_n)$if and only if it has the Cayley-Bacharach property and the Kähler different is non-zero at a certain degree. When$\mathbb{X}$has the$(\star)$-property, we characterize the Cayley-Bacharach property of$\mathbb{X}$in terms of its components under the canonical projections. Show More • ### Published online May 4, 2022 #### Complete characterization of odd factors via the size, spectral radius or distance spectral radius of graphs Shuchao Li and Shujing Miao Abstract : Given a graph$G,$a$\{1,3,\ldots,2n-1\}$-factor of$G$is a spanning subgraph of$G$, in which each degree of vertices is one of$\{1,3,\ldots,2n-1\}$, where$n$is a positive integer. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of$G$to guarantee that$G$contains a$\{1,3,\ldots,2n-1\}$-factor. Then we determine an upper bound on the distance spectral radius of$G$to ensure that$G$has a$\{1,3,\ldots,2n-1\}$-factor. Furthermore, we construct some extremal graphs to show all the bounds obtained in this contribution are best possible. • ### Published online May 4, 2022 #### Basic formulas for the double integral transform of functionals on abstract Wiener space Hyun Soo Chung Abstract : In this paper, we establish several basic formulas among the double-integral transforms, the double-convolution products, and the inverse double-integral transforms of cylinder functionals on abstract Wiener space. We then discuss possible relationships involving the double-integral transform. ## Current Issue ### March, 2022 Vol.59 No.2 ## Most Read • ### On the existence of Graham partitions with congruence conditions Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park Bull. Korean Math. Soc. 2022; 59(1): 15-25 https://doi.org/10.4134/BKMS.b200730 • ### On weighted Browder spectrum Preeti Dharmarha, Sarita Kumari Bull. Korean Math. Soc. 2022; 59(1): 1-13 https://doi.org/10.4134/BKMS.b200328 • ### Admissible balanced pairs over formal triangular matrix rings Lixin Mao Bull. Korean Math. Soc. 2021; 58(6): 1387-1400 https://doi.org/10.4134/BKMS.b200924 • ### One-sided fattening of the graph in the real projective plane Jaeyoo Choy, Hahng-Yun Chu Bull. Korean Math. Soc. 2022; 59(1): 27-43 https://doi.org/10.4134/BKMS.b201080 ## Most Downloaded • ### On weighted Browder spectrum Preeti Dharmarha, Sarita Kumari Bull. Korean Math. Soc. 2022; 59(1): 1-13 https://doi.org/10.4134/BKMS.b200328 • ### On the existence of Graham partitions with congruence conditions Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Sang June Lee, Poo-Sung Park Bull. Korean Math. Soc. 2022; 59(1): 15-25 https://doi.org/10.4134/BKMS.b200730 • ### Entire solutions of differential-difference equations of Fermat type Peichu Hu, Wenbo Wang, Linlin Wu Bull. Korean Math. Soc. 2022; 59(1): 83-99 https://doi.org/10.4134/BKMS.b210099 • ### Sasakian 3-Metric as a$\ast\$-Conformal Ricci Soliton Represents a Berger Sphere

Dibakar Dey

Bull. Korean Math. Soc. 2022; 59(1): 101-110
https://doi.org/10.4134/BKMS.b210125