Communications of the
Korean Mathematical Society
CKMS
ISSN(Print) 1225-1763
ISSN(Online) 2234-3024
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2023-04-30
A characterization of zero divisors and topological divisors of zero in $C[a,b]$ and $\ell^\infty$
Harish Chandra, Anurag Kumar PatelAbstract : We give a characterization of zero divisors of the ring $C[a,b]$. Using the Weierstrass approximation theorem, we completely characterize topological divisors of zero of the Banach algebra $C[a,b]$. We also characterize the zero divisors and topological divisors of zero in $\ell^\infty$. Further, we show that zero is the only zero divisor in the disk algebra $\mathscr{A}(\mathbb{D})$ and that the class of singular elements in $\mathscr{A}(\mathbb{D})$ properly contains the class of topological divisors of zero. Lastly, we construct a class of topological divisors of zero of $\mathscr{A}(\mathbb{D})$ which are not zero divisors.
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2023-10-31
Transversal lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds
Shiv Sharma Shukla, Vipul SinghAbstract : In this paper, we introduce and study two new classes of lightlike submersions, called radical transversal and transversal lightlike submersions between an indefinite Sasakian manifold and a lightlike manifold. We give examples and investigate the geometry of distributions involved in the definitions of these lightlike submersions. We also study radical transversal and transversal lightlike submersions from an indefinite Sasakian manifold onto a lightlike manifold with totally contact umbilical fibers.
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2023-04-30
Unitary analogues of a generalized number-theoretic sum
Traiwat Intarawong, Boonrod YuttananAbstract : In this paper, we investigate the sums of the elements in the finite set $\{x^{k}:1\leq x\leq\frac{n}{m},\gcd_u(x,n)=1\}$, where $k$, $m$ and $n$ are positive integers and $\gcd_u(x,n)$ is the unitary greatest common divisor of $x$ and $n$. Moreover, for some cases of $k$ and $m$, we can give the explicit formulae for the sums involving some well-known arithmetic functions.
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2023-04-30
A simple proof for a result on $n$-Jordan homomorphisms
Choonkil Park, Abbas Zivari-KazempourAbstract : In this short note, we give a simple proof of the main theorem of \cite{Cheshmavar} which states that every $n$-Jordan homomorphism $h:A\longrightarrow B$ between two commutative algebras $A$ and $B$ is an $n$-homomorphism.
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2024-04-30
A weaker notion of the finite factorization property
Henry Jiang, Shihan Kanungo, Hwisoo KimAbstract : An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid $M$ is a length-finite factorization monoid if each $b \in M$ has only finitely many factorizations of any prescribed length. An additive submonoid of $\mathbb{R}_{\ge 0}$ is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.
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2023-07-31
Hyers-Ulam stability of fractional stochastic differential equations with random impulse
Dumitru Baleanu, Banupriya Kandasamy, Ramkumar Kasinathan, Ravikumar Kasinathan, Varshini SandrasekaranAbstract : The goal of this study is to derive a class of random impulsive non-local fractional stochastic differential equations with finite delay that are of Caputo-type. Through certain constraints, the existence of the mild solution of the aforementioned system are acquired by Kransnoselskii's fixed point theorem. Furthermore through Ito isometry and Gronwall's inequality, the Hyers-Ulam stability of the reckoned system is evaluated using Lipschitz condition.
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2023-07-31
The dimension graph for modules over commutative rings
Shiroyeh PayroviAbstract : Let $R$ be a commutative ring and $M$ be an $R$-module. The dimension graph of $M$, denoted by $DG(M)$, is a simple undirected graph whose vertex set is $Z(M)\setminus {\rm Ann}(M)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\dim M/(x, y)M=\min\{\dim M/xM, \dim M/yM\}$. It is shown that $DG(M)$ is a disconnected graph if and only if (i) ${\rm Ass}(M)=\{\mathfrak p, \mathfrak q\}$, $Z(M)=\mathfrak p\cup \mathfrak q$ and ${\rm Ann}(M)=\mathfrak p\cap \mathfrak q$. (ii) $\dim M=\dim R/\mathfrak p=\dim R/\mathfrak q$. (iii) $\dim M/xM=\dim M$ for all $x\in Z(M)\setminus {\rm Ann}(M)$. Furthermore, it is shown that ${\rm diam}(DG(M))\leq 2$ and ${\rm gr}({DG(M)})=3$, whenever $M$ is Noetherian with $|Z(M)\setminus {\rm Ann}(M)| \geq 3$ and $DG(M)$ is a connected graph.
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2023-04-30
Uniqueness of meromorphic function with its $k$-th derivative sharing two small functions under different weights
Abhijit Banerjee, Arpita KunduAbstract : In the paper, we have exhaustively studied about the uniqueness of meromorphic function sharing two small functions with its $k$-th derivative as these types of results have never been studied earlier. We have obtained a series of results which will improve and extend some recent results of Banerjee-Maity \cite{Ban-Maity_Contemp.}.
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2023-04-30
On graded $J$-ideals over graded rings
Tamem Al-Shorman, Malik Bataineh, Ece Yetkin CelikelAbstract : The goal of this article is to present the graded $J$-ideals of $G$-graded rings which are extensions of $J$-ideals of commutative rings. A graded ideal $P$ of a $G$-graded ring $R$ is a graded $J$-ideal if whenever $x,y\in h(R)$, if $xy\in P$ and $x\not\in J(R)$, then $y\in P$, where $h(R)$ and $J(R)$ denote the set of all homogeneous elements and the Jacobson radical of $R$, respectively. Several characterizations and properties with supporting examples of the concept of graded $J$-ideals of graded rings are investigated.
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2023-04-30
Simple formulations on circulant matrices with alternating Fibonacci
Sugi GuritmanAbstract : In this article, an alternating Fibonacci sequence is defined from a second-order linear homogeneous recurrence relation with constant coefficients. Then, the determinant, inverse, and eigenvalues of the circulant matrices with entries in the first row having the formation of the sequence are formulated explicitly in a simple way. In this study, the method for deriving the formulation of the determinant and inverse is simply using traditional elementary row or column operations. For the eigenvalues, the known formulation from the case of general circulant matrices is simplified by considering the specialty of the sequence and using cyclic group properties. We also propose algorithms for the formulation to show how efficient the computations are.
Most Read
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A characterization of zero divisors and topological divisors of zero in $C[a,b]$ and $\ell^\infty$
Harish Chandra, Anurag Kumar Patel
Commun. Korean Math. Soc. 2023; 38(2): 451-459
https://doi.org/10.4134/CKMS.c220108 -
Transversal lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds
Shiv Sharma Shukla, Vipul Singh
Commun. Korean Math. Soc. 2023; 38(4): 1191-1213
https://doi.org/10.4134/CKMS.c220309 -
Unitary analogues of a generalized number-theoretic sum
Traiwat Intarawong, Boonrod Yuttanan
Commun. Korean Math. Soc. 2023; 38(2): 355-364
https://doi.org/10.4134/CKMS.c220139 -
A simple proof for a result on $n$-Jordan homomorphisms
Choonkil Park, Abbas Zivari-Kazempour
Commun. Korean Math. Soc. 2023; 38(2): 487-490
https://doi.org/10.4134/CKMS.c220136
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A study of differential identities on $\sigma$-prime rings
Adnan Abbasi, Md Arshad Madni, Muzibur Rahman Mozumder
Commun. Korean Math. Soc. 2023; 38(3): 679-693
https://doi.org/10.4134/CKMS.c220240 -
Unitary analogues of a generalized number-theoretic sum
Traiwat Intarawong, Boonrod Yuttanan
Commun. Korean Math. Soc. 2023; 38(2): 355-364
https://doi.org/10.4134/CKMS.c220139 -
The dimension graph for modules over commutative rings
Shiroyeh Payrovi
Commun. Korean Math. Soc. 2023; 38(3): 733-740
https://doi.org/10.4134/CKMS.c220273 -
Hyers-Ulam stability of fractional stochastic differential equations with random impulse
Dumitru Baleanu, Banupriya Kandasamy, Ramkumar Kasinathan, Ravikumar Kasinathan, Varshini Sandrasekaran
Commun. Korean Math. Soc. 2023; 38(3): 967-982
https://doi.org/10.4134/CKMS.c220231
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