Communications of the
Korean Mathematical Society
CKMS
ISSN(Print) 1225-1763
ISSN(Online) 2234-3024
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2023-07-31
Local-global principle and generalized local cohomology modules
Bui Thi Hong Cam, Nguyen Minh Tri, Do Ngoc YenAbstract : Let $\mathcal{M}$ be a stable Serre subcategory of the category of $R$-modules. We introduce the concept of $\mathcal{M}$-minimax $R$-modules and investigate the local-global principle for generalized local cohomology modules that concerns to the $\mathcal{M}$-minimaxness. We also provide the $\mathcal{M}$-finiteness dimension $f^{\mathcal{M}}_I(M,N)$ of $M,N$ relative to $I$ which is an extension the finiteness dimension $f_I(N)$ of a finitely generated $R$-module $N$ relative to $I$.
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2023-10-31
On graded $N$-irreducible ideals of commutative graded rings
Anass Assarrar, Najib MahdouAbstract : Let $R$ be a commutative graded ring with nonzero identity and $n$ a positive integer. Our principal aim in this paper is to introduce and study the notions of graded $n$-irreducible and strongly graded $n$-irreducible ideals which are generalizations of $n$-irreducible and strongly $n$-irreducible ideals to the context of graded rings, respectively. A proper graded ideal $I$ of $R$ is called graded $n$-irreducible (respectively, strongly graded $n$-irreducible) if for each graded ideals $I_{1}, \ldots,I_{n+1}$ of $R$, $I=I_{1} \cap \cdots \cap I_{n+1}$ (respectively, $I_{1} \cap \cdots \cap I_{n+1} \subseteq I$ ) implies that there are $n$ of the $I_{i}$ 's whose intersection is $I$ (respectively, whose intersection is in $I$). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded $n$-irreducible ideal which is not an $n$-irreducible ideal and an example of a graded ideal which is graded $n$-irreducible, but not graded $(n-1)$-irreducible.
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2023-07-31
A note on certain transformation formulas related to Appell, Horn and Kamp\'{e} de F\'{e}riet functions
Asmaa Orabi Mohammed, Medhat Ahmed Rakha, Arjun K. RathieAbstract : In 2019, Mathur and Solanki \cite{7,8} obtained a few transformation formulas for Appell, Horn and the Kamp\'{e} de F\'{e}riet functions. Unfortunately, some of the results are well-known and very old results in literature while others are erroneous. Thus the aim of this note is to provide the results in corrected forms and some of the results have been written in more compact form.
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2023-10-31
On graded $(m, n)$-closed submodules
Rezvan VarmazyarAbstract : Let $A$ be a $G$-graded commutative ring with identity and $M$ a graded $A$-module. Let $m, n$ be positive integers with $m>n$. A proper graded submodule $L$ of $M$ is said to be graded $(m, n)$-closed if $a^{m}_g\cdot x_t\in L$ implies that $a^{n}_g\cdot x_t\in L$, where $a_g\in h(A)$ and $x_t\in h(M)$. The aim of this paper is to explore some basic properties of these class of submodules which are a generalization of graded $(m, n)$-closed ideals. Also, we investigate $GC^{m}_n-rad$ property for graded submodules.
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2022-10-31
Generalized Padovan sequences
Jhon J. Bravo, Jose L. HerreraAbstract : The Padovan sequence is the third-order linear recurrence $(\mathcal{P}_n)_{n\geq 0}$ defined by $\mathcal{P}_n=\mathcal{P}_{n-2}+\mathcal{P}_{n-3}$ for all $n\geq 3$ with initial conditions $\mathcal{P}_0=0$ and $\mathcal{P}_1=\mathcal{P}_2=1$. In this paper, we investigate a generalization of the Padovan sequence called the $k$-generalized Padovan sequence which is generated by a linear recurrence sequence of order $k\geq 3$. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences.
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2024-01-31
On strongly quasi $J$-ideals of commutative rings
El Mehdi Bouba
, Yassine EL-khabchi, Mohammed Tamekkante Abstract : Let $R$ be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi $J$-ideals lying properly between the class of $J$-ideals and the class of quasi $J$-ideals. A proper ideal $I$ of $R$ is called a strongly quasi $J$-ideal if, whenever $a$, $b\in R$ and $ab\in I$, then $a^{2}\in I$ or $b\in {\rm Jac}(R)$. Firstly, we investigate some basic properties of strongly quasi $J$-ideals. Hence, we give the necessary and sufficient conditions for a ring $R$ to contain a strongly quasi $J$-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi $J$-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.
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2024-01-31
Nonnil-$S$-coherent rings
Najib Mahdou, El Houssaine OubouhouAbstract : Let $R$ be a commutative ring with identity. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $\phi$-ring. Let $R$ be a $\phi$-ring and $S$ be a multiplicative subset of $R$. In this paper, we introduce and study the class of nonnil-$S$-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are $S$-finitely presented. Also, we define the concept of $\phi$-$S$-coherent rings. Among other results, we investigate the $S$-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-$S$-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.
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2023-07-31
S-coherent property in trivial extension and in amalgamated duplication
MOHAMED CHHITI, SALAH EDDINE MAHDOUAbstract : Bennis and El Hajoui have defined a (commutative unital) ring $R$ to be $S$-coherent if each finitely generated ideal of $R$ is a $S$-finitely presented $R$-module. Any coherent ring is an $S$-coherent ring. Several examples of $S$-coherent rings that are not coherent rings are obtained as byproducts of our study of the transfer of the $S$-coherent property to trivial ring extensions and amalgamated duplications.
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2023-07-31
The $p$-part of divisor class numbers for cyclotomic function fields
Daisuke ShiomiAbstract : In this paper, we construct explicitly an infinite family of primes $P$ with $h_P^{\pm} \equiv 0 \pmod {q^{\deg P}}$, where $h_P^{\pm}$ are the plus and minus parts of the divisor class number of the $P$-th cyclotomic function field over $\mathbb{F}_q(T)$. By using this result and Dirichlet's theorem, we give a condition of $A, M \in \mathbb{F}_q[T]$ such that there are infinitely many primes $P$ satisfying with $h_P^{\pm} \equiv 0 \pmod {p^e}$ and $P \equiv A \pmod M$.
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2023-10-31
The dimension of the maximal spectrum of some ring extensions
Rachida EL KHALFAOUI, Najib MahdouAbstract : Let $A$ be a ring and $\mathcal{J} = \{\text{ideals $I$ of $A$} \,|\, J(I) = I\}$. The Krull dimension of $A$, written $\dim A$, is the sup of the lengths of chains of prime ideals of $A$; whereas the dimension of the maximal spectrum, denoted by $\dim_\mathcal{J} A$, is the sup of the lengths of chains of prime ideals from $\mathcal{J}$. Then $\dim_{\mathcal{J}} A\leq \dim A$. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property $J$-Noetherian to ring extensions.
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Pascal's hexagon theorem reproved by elementary tools only
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Functions subordinate to the exponential function
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Choonkil Park, Abbas Zivari-Kazempour
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The measurability of Hewitt-Stromberg measures and dimensions
Zied Douzi, Bilel Selmi, Haythem Zyoudi
Commun. Korean Math. Soc. 2023; 38(2): 491-507
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Oriented transformations on a finite chain: another description
Vítor Hugo Fernandes
Commun. Korean Math. Soc. 2023; 38(3): 725-731
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Fractional integration and differentiation of the $(p,q)$--extended modified Bessel function of the second kind and integral transforms
Purnima Chopra, Mamta Gupta, Kanak Modi
Commun. Korean Math. Soc. 2023; 38(3): 755-772
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Some classes of operators related to $(m, n)$-paranormal and $(m, n)^*$-paranormal operators
SHINE LAL ENOSE, RAMYA PERUMAL, PRASAD THANKARAJAN
Commun. Korean Math. Soc. 2023; 38(4): 1075-1090
https://doi.org/10.4134/CKMS.c220355
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