Abstract : Let $R$ be a commutative ring with identity. Let $R$ be an integral domain and $M$ a torsion-free $R$-module. We investigate the relation between the notion of e-exactness, recently introduced by Akray and Zebari \cite{a a}, and generalized the concept of homology, and establish a relation between e-exact sequences and homology of modules. We modify some applications of e-exact sequences in homology and reprove some results of homology with e-exact sequences such as horseshoe lemma, long exact sequences, connecting homomorphisms and etc. Next, we generalize two special drived functor $Tor$ and $Ext$, and study some properties of them.
Abstract : The main goal of the paper is the introduction of the notion of conformal hemi-slant submersions from almost contact metric manifolds onto Riemannian manifolds. It is a generalization of conformal anti-invariant submersions, conformal semi-invariant submersions and conformal slant submersions. Our main focus is conformal hemi-slant submersion from cosymplectic manifolds. We tend also study the integrability of the distributions involved in the definition of the submersions and the geometry of their leaves. Moreover, we get necessary and sufficient conditions for these submersions to be totally geodesic, and provide some representative examples of conformal hemi-slant submersions.
Abstract : In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus $T$. In particular, we introduce the notion of {\it mixed knotoids} in $S^2$, that generalizes the notion of mixed links in $S^3$, and we present an isotopy theorem for mixed knotoids. We then generalize the Kauffman bracket polynomial, $$, for mixed knotoids and we present a state sum formula for $$. We also introduce the notion of {\it mixed pseudo knotoids}, that is, multi-knotoids on two components with some missing crossing information. More precisely, we present an isotopy theorem for mixed pseudo knotoids and we extend the Kauffman bracket polynomial for pseudo mixed knotoids. Finally, we introduce the theories of {\it mixed braidoids} and {\it mixed pseudo braidoids} as counterpart theories of mixed knotoids and mixed pseudo knotoids, respectively. With the use of the $L$-moves, that we also introduce here for mixed braidoid equivalence, we formulate and prove the analogue of the Alexander and the Markov theorems for mixed knotoids. We also formulate and prove the analogue of the Alexander theorem for mixed pseudo knotoids.
Abstract : 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.
Abstract : An le-module $M$ over a commutative ring $R$ is a complete lattice ordered additive monoid $(M, \leqslant, +)$ having the greatest element $e$ together with a module like action of $R$. This article characterizes the le-modules $_RM$ such that the pseudo-prime spectrum $X_M$ endowed with the Zariski topology is a Noetherian topological space. If the ring $R$ is Noetherian and the pseudo-prime radical of every submodule elements of $_{R}M$ coincides with its Zariski radical, then $X_{M}$ is a Noetherian topological space. Also we prove that if $R$ is Noetherian and for every submodule element $n$ of $M$ there is an ideal $I$ of $R$ such that $V(n) = V(Ie)$, then the topological space $X_{M}$ is spectral.
Abstract : 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.
Abstract : Recently, Bre\v{s}ar's Jordan $\{g,h\}$-derivations have been investigated on triangular algebras. As a first aim of this paper, we extend this study to an interesting general context. Namely, we introduce the notion of Jordan $\mathcal{G}_n$-derivations, with $n \ge 2$, which is a natural generalization of Jordan $\{g,h\}$-derivations. Then, we study this notion on path algebras. We prove that, when $n > 2$, every Jordan $\mathcal{G}_n$-derivation on a path algebra is a $\{g,h\}$-derivation. However, when $n = 2$, we give an example showing that this implication does not hold true in general. So, we characterize when it holds. As a second aim, we give a positive answer to a variant of Lvov-Kaplansky conjecture on path algebras. Namely, we show that the set of values of a multi-linear polynomial on a path algebra $KE$ is either $\{0\}$, $KE$ or the space spanned by paths of a length greater than or equal to $1$.
Abstract : Let $\mathcal{H}_0$ be the set of rings $R$ such that $Nil(R) = Z(R)$ is a divided prime ideal of $R$. The concept of maximal non $\phi$-chained subrings is a generalization of maximal non valuation subrings from domains to rings in $\mathcal{H}_0$. This generalization was introduced in \cite{rahul} where the authors proved that if $R \in \mathcal{H}_0$ is an integrally closed ring with finite Krull dimension, then $R$ is a maximal non $\phi$-chained subring of $T(R)$ if and only if $R$ is not local and $|[R, T(R)]|$ = $\dim (R) + 3$. This motivates us to investigate the other natural numbers $n$ for which $R$ is a maximal non $\phi$-chained subring of some overring $S$. The existence of such an overring $S$ of $R$ is shown for $3\leq n \leq 6$, and no such overring exists for $n = 7$.
Abstract : Special functions and Geometric function theory are close related to each other due to the surprise use of hypergeometric function in the solution of the Bieberbach conjecture. The purpose of this paper is to provide a set of sufficient conditions under which the normalized four parametric Wright function has lower bounds for the ratios to its partial sums and as well as for their derivatives. The sufficient conditions are also obtained by using Alexander transform. The results of this paper are generalized and also improved the work of M. Din et al.~cite{din}. Some examples are also discussed for the sake of better understanding of this article.
Abstract : 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.
Ismael Akray, Amin Mahamad Zebari
Commun. Korean Math. Soc. 2023; 38(1): 21-38
https://doi.org/10.4134/CKMS.c210349
Vinay Kumar, Rajendra Prasad, Sandeep Kumar Verma
Commun. Korean Math. Soc. 2023; 38(1): 205-221
https://doi.org/10.4134/CKMS.c210433
Ioannis Diamantis
Commun. Korean Math. Soc. 2022; 37(4): 1221-1248
https://doi.org/10.4134/CKMS.c210169
Harish Chandra, Anurag Kumar Patel
Commun. Korean Math. Soc. 2023; 38(2): 451-459
https://doi.org/10.4134/CKMS.c220108
Vinay Kumar, Rajendra Prasad, Sandeep Kumar Verma
Commun. Korean Math. Soc. 2023; 38(1): 205-221
https://doi.org/10.4134/CKMS.c210433
Ismael Akray, Amin Mahamad Zebari
Commun. Korean Math. Soc. 2023; 38(1): 21-38
https://doi.org/10.4134/CKMS.c210349
Anjan Kumar Bhuniya, Manas Kumbhakar
Commun. Korean Math. Soc. 2023; 38(1): 1-9
https://doi.org/10.4134/CKMS.c210057
Traiwat Intarawong, Boonrod Yuttanan
Commun. Korean Math. Soc. 2023; 38(2): 355-364
https://doi.org/10.4134/CKMS.c220139
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