Let R be a commutative ring with unity and M an R-module. We say that M satisfies accr if the ascending chain of submodules of the form N : B ⊆ N : B² ⊆ ⋯ terminates for every submodule N of M and every finitely generated ideal B of R. The ring R satisfies the accr condition if the R-module R does. In this paper, we give a necessary and sufficient condition for amalgamated algebras A ⨝^f J (respectively, the trivial extension of R by M, R(+)M) to be an accr ring. In the second part of this paper, we introduce the concept of uniformly S-accr modules and study its properties. Let S be a multiplicative subset of R. We say that M satisfies uniformly S-accr if there exists an s ∈ S such that for every submodule N of M and every finitely generated ideal B of R the ascending sequence of the form N : B ⊆ N : B² ⊆ ⋯ is stationary with respective to s. We study basic results of uniformly S-accr ring. Also, we give an S-version of several well-known result about accr rings. The concept of Nagata's idealization R(+)M and the amalgamated algebras A ⨝^f J is studied.

Let R be a commutative ring with unity and M be an R-module. The prime submodule sum graph of M, denoted by P SS(M), is an undirected graph whose vertices are nonzero proper submodules of M and two distinct vertices N and P are adjacent if and only if N + P is a prime submodule of M. In this paper, we study some basic graph theoretic properties of the graph P SS(M). We also discuss some results on homomorphisms of P SS(M). Finally, planarity and domination numbers of P SS(M) are also investigated.

In this paper, we introduce a new broad generalization of prime ideals in commutative rings using endomorphisms defined on them. Let R be a commutative ring with 1 ≠ 0, and let f : R → R be a ring endomorphism. An ideal I of R is called an f-prime ideal if, whenever a, b ∈ R and ab ∈ I, then a ∈ I or f(b) ∈ I. Firstly, we study various basic properties of f-prime ideals. We show that this type of ideals can be useful in detecting the prime property of ideals in commutative rings. In particular, we give some new necessary and sufficient conditions for an ideal to be a prime ideal in terms of f-prime ideals. We prove through examples and results that these ideals enjoy some exotic features that prime ideals lack. In particular, we fully characterize rings that admit f-prime ideals, rings in which only (0) is an f-prime ideal, and rings in which every ideal is an f-prime ideal. Finally, we relate f-prime ideals of a ring R with other such ideals of various algebraic constructions on R.

In this paper we introduce a class of graded Poisson color algebras as a natural generalization of graded Poisson algebras and graded Poisson superalgebras. For Λ an arbitrary abelian group, we show that any of such Λ-graded Poisson color algebra 𝒫, with a symmetric Λ-support is of the form 𝒫 = 𝒰 ⊕ Σ_j I_j, with 𝒰 a subspace of 𝒫₁ and any I_j a well described graded ideal of 𝒫, satisfying {I_j, I_k} + I_jI_k = 0 if j ≠ i. Furthermore, under certain conditions, the gr-simplicity of 𝒫 is characterized and it is shown that 𝒫 is the direct sum of the family of its graded simple ideals.

Let S be a semigroup. In this paper we find the solutions f, g, h : S ⟶ ℂ of the Cosine-Sine other functional equation f(xy) = f(x)f(y) + g(x)g(y) + h(x)h(y), x, y ∈ S, in terms of multiplicative functions and the solutions of special cases of the sine and the cosine-sine functional equations. As an application, we give the solutions f, g, h : S ⟶ ℂ of the functional equation f(xy) = g(x)h(y) + h(x)g(y) + f(x)f(y), x, y ∈ S.

Let R be a commutative ring with identity and 𝓙 be an ideal of R. The aim of this paper is to introduce and study a new class of rings that is closely related to the class of Artinian rings. An ideal I of R is said to be a 𝓙-ideal if I ⊈ 𝓙. We say that R is 𝓙-Artinian if every descending chain of 𝓙-ideals is stationary. We show that the theories of 𝓙-Artinian rings resemble that of Artinian rings. We also provide several characterizations of 𝓙-Artinian rings in terms of Artinian rings. Furthermore, we obtain some connections of 𝓙-Artinian rings with semisimple rings, 𝜙-von Neumann regular rings, trivial ring extensions, and zero-dimensional rings.

In this paper, we extend the concepts of Noetherian and coherent modules to ℵ₀-Noetherian and ℵ₀-coherent modules. A module is defined as ℵ₀-Noetherian if every submodule is countably generated, and ℵ₀-coherent if it is countably generated and every countably generated submodule is countably presented. We investigate the fundamental properties of these modules and explore their behavior under several constructions, such as trivial ring extensions, fiber products, and amalgamations. Additionally, we demonstrate that ℵ₀-Noetherian and ℵ₀-coherent modules are closed under localization and provide new insights into the structure of these modules in comparison to their classical counterparts.

Every numerical semigroup can be expressed as the intersection of irreducible numerical semigroups. We show that the unions of sets of lengths of factorizations of numerical semigroups into irreducible numerical semigroups are all equal to ℕ_≥2.

Let R be a commutative Noetherian ring and M be an R-module such that, for each submodule L of M, the set of associated prime ideals of the quotient module M/L is finite. In this paper, it is shown that M admits a finitely generated submodule N such that the set of associated prime ideals of M/N and the support of M/N are equal.

In this study, we use the Laguerre collocation technique to approximate the solution of the higher-order Caputo fractional differential equation with multi-terms and initial conditions. This process converts the provided differential equation into an algebraic one, and the unknown coefficients of the problem's Laguerre series solution are determined. The error boundaries are evaluated for numerical calculations to ensure the method's accuracy. An example is given to demonstrate the established technique and use mathematical computing techniques to assess how an approximation solution behaves.

In this article, we introduce the class of generalized (h, m)-exponentially convex functions on fractal sets (ℝ^α, 0 < α ≤ 1), an extension of the category of generalized (h - m)-convex functions and consequently prove new Hermite-Hadamard type and Fejér-Hermite-Hadamard type integral properties of these mappings. Additionally, we deduce specific inequalities concerning h-convexity and exponential (h, m)-convexity, which arise as particular cases in our analysis. To verify the accuracy of the current findings, we also examine some novel applications for generalized special means.

The exploration in this paper captures several closure and inclusion properties of diverse forms of analytic-univalent function f defined in the unit disk. The explored set ${\mathfrak{S}}^*_{q,t}(n,\,b,\,u;\,{\gamma},\,{\alpha},\,{\beta})$ generalizes the class of Janowski-q-starlike functions defined in connection with the Jackson's differentiation, the famous Opoola q-differential operator and the principle of subordination. Successively, the set consists of several known and new subsets of the univalently-starlike functions. The results show that the arithmetic mean, weighted mean, convex linear combination, convolution of functions f and h; and several known (q-)integral operators are closed in the set. Likewise, two novel q-analogue of λ-neighborhood inclusion properties are introduced and proved for the set ${\mathfrak{S}}^*_{q,t}(n,\,b,\,u;\,{\gamma},\,{\alpha},\,{\beta})$.

Present study explores the extension of hypergeometric functions using matrix arguments, focusing on the Gauss and Wright-type generalized hypergeometric functions viz. ₂F₁, ₂R₁, _pR_q, _pψ_q. Using foundational concepts like Pochhammer symbol and gamma function of matrix arguments, analogues of above functions have been introduced in matrix arguments using zonal polynomials, and several properties of which have been studied, contributing to the development in the theory of special functions, opening pathways for new research.

In the current article, we introduce a new fashion of a multi-additive mapping, as a d-dimensional multi-additive mapping and present a characterization of its structure (unified as an equation). The main aim of this study is to establish some stability results of the d-dimensional multi-additive mappings by applying the so-called direct (Hyers) method in the setting of neutrosophic Banach spaces.

If 0 < p ≤ 1, 𝜔 is a weight on ℤ², 𝒜 is a unital complex Banach algebra and if f is a continuous 𝒜 valued function on 𝕋² such that ${\sum{_{{(m,n)}{\in}{\mathbb{Z}}^2}}\,{\parallel}{\hat{f}}(m,n){\parallel}^p{\omega}(m,n)\,<\,{\infty}$, where ${\hat{f}}(m,n)$ are Fourier coefficients of f, and f(z, w) is left invertible in 𝒜 for all (z, w) ∈ 𝕋², then it is shown that there is a weight 𝜈 on ℤ² and a continuous function g : 𝕋₂ → 𝒜 such that 1 ≤ 𝜈 ≤ 𝜔, 𝜈 is constant if and only if 𝜔 is constant, ${\sum{_{{(m,n)}{\in}{\mathbb{Z}}^2}}\,{\parallel}{\hat{g}}(m,n){\parallel}^p{\omega}(m,n)\,<\,{\infty}$ and g is a left inverse of f. A similar result is obtained for a continuous function f from 𝕋^∞ (the countable copies of 𝕋) to 𝒜.

In this note, we prove that if pairwise distances are preserved for n-points in the unit disc, then there exists a holomorphic or anti-holomorphic interpolation function from the unit disc to itself.

In this article, we have successfully achieved hybrid projective synchronization between chaotic systems of different dimensions. By applying Lyapunov stability theory, we have designed appropriate nonlinear controllers to realize both increased and reduced order hybrid projective synchronization. As a demonstration of this synchronization approach, we used recently developed 5D and 4D hyperchaotic systems. Numerical simulations and graphical representations were performed using MATLAB, showing a strong correlation between theoretical predictions and analytical results.