The concepts have been introduced in Almost DistributiveLattices(ADLs), namely, R-ideals and-ideals. A set ofconditions has been identified that are equivalent to converting anE-ideal into an R-ideal. Moreover, it has been derived that forany E-ideal, there exists a homomorphism with a dual dense kernel,which is itself an R-ideal. The characterization of-ideals interms of R-ideals and congruences has been established. Additionally,equivalent conditions have been established to demonstratethat the space of all prime-ideals forms a Hausdorff space.
We study those modules in which all submodules contain a nonzero submodule with Noetherian dimension less than or equal to $\alpha $ where $\alpha $ is the least ordinal number with this property, calling them modules that have enough $\alpha$-noetherians. Also, we study those modules in which any nonzero factor module contains a nonzero submodule with Noetherian dimension less than or equal to $\alpha$, where $\alpha$ is the least ordinal number with this property, calling them $\alpha$-seminoetherians. Our work extends the results of Kourki and Tribak in Commu. Algebra (2022).
The rings considered in this paper are commutative with identity and the modules considered are modules over commutative rings and are unitary. We use R to denote a ring, S to denote a multiplicatively closed subset of R, and M to denote a module over R. We say that M is an S-Laskerian (respectively, a strongly S-Laskerian) R-module if M is an S-finite R-module and for any submodule N of M, either (N: M) meets S or there exist t in S and an S-decomposable (respectively, a strongly S-decomposable) submodule K of M such that tN is a submodule of K and K is contained in N. We say that M is a weakly S-Laskerian (respectively, weakly strongly S-Laskerian) R-module if each S-finite proper submodule of M is an S-Laskerian (respectively, a strongly Laskerian ) R-module. In this paper, we discuss some results on basic properties of weaklly S-Laskerian modules and we extend some of the properties of S-Laskerian modules to weakly S-Laskerian modules.
In this paper, we consider certain classes of hypergraphs constructed from the tensor join of hypergraphs, specifically the tensor join of hypergraphs constrained by vertex subsets and the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. We determine some eigenvalues of the adjacency tensor of these classes of hypergraphs by establishing corresponding eigenvectors. We demonstrate that the eigenvalues of the adjacency tensor of the constituting hypergraphs are also eigenvalues of the adjacency tensor of the join of a set of hypergraphs. Furthermore, as a special case of our results, we provide some eigenvalues and eigenvectors of the adjacency tensor for the join of non-uniform hypergraphs on a backbone hypergraph $H$ (and, similarly, for the join of $m$-uniform hypergraphs on a backbone hypergraph $H$). Additionally, we establish a relationship between the eigenvalues of the adjacency tensor of a hypergraph $H$ and certain eigenvalues of the adjacency tensor of the $(H, \mathcal{T}_{\mathcal{S}})$-join of hypergraphs constrained by $\mathcal{S}$. Using this relationship, we determine some eigenvalues and their corresponding eigenvectors for the adjacency tensor of the lexicographic product of two hypergraphs.
In their construction of the Stone-\v{C}ech compactification using only ring ideals (as opposed to $\ell$-ideals), Banaschewski and Sioen define a certain nucleus on the coherent frame $\Rid(A)$ of radical ideals of a commutative ring $A$. In this paper we extend this nucleus to any algebraic frame that has a dense compact element and in which the meet of two compact elements is compact. Of course, not every such algebraic frame is coherent, so the extension is indeed a genuine extension. We then study some properties of this nucleus which are not considered by Banaschewski and Sioen.
In this paper, the concept of non-parallel Krull (resp., Noetherian) dimension of an R-module is introduced, and some related properties are investigated. Using these concepts, we generalize some of the results of our previous work and try to obtain some appropriate new results about np-Artinian (resp., np-Noetherian) modules. We give a characterization for modules with non-parallel Krull (resp., Noetherian) dimension and show that these modules have finite type dimension. It is shown that any R-module M with non-parallel Krull dimension at most cx is either atomic or lambda-f.e. for some lambda < alpha. Also, it is proved that any np-Noetherian R-module has non-parallel Krull dimension. In particular, for semiprime right non-atomic rings, we show that Krull dimension and non-parallel Krull dimension coincide.
This paper introduces the concept of k-hyperideals of ternary hypersemirings and explores some of its fundamental characteristics. Next, k-hyperideals are used to characterize prime ternary hypersemirings and ternary hypersemifields. Moreover, k-hyperideals explain the quotient of ternary hypersemirings. Finally, the new kind of ternary hypersemirings is formed using regular equivalence relations, and the set of all k-hyperideals is generated from them.
Let D be a division ring over its center F. Let GL(n)(D) be the general linear group over D. In this article we prove that if GL(n)(D) contains a torsion maximal subgroup M, then D = F , charF = p > 0 and F is algebraic over its prime subfield F-p whenever one of the following conditions occurs: (1) D is algebraic over F; (2) there exists an element a E M such that C-Mn(D)(F [a]) is algebraic over F.
Let X be a Tychonoff space and R[X] be the collection of all subrings of C(X) that separate points and contain the identity element 1. In this paper, we establish a correspondence between the ideals in A(X) is an element of R[X] and the z(A)(gamma)-filters on the completion of X with respect to a uniform structure arising from the functions in A(X). We also explore some properties of z-ideals, z(A)-ideals and maximal ideals in these types of subrings of C(X). For each subset A(X) of C(X), let [A(X)]c= {f +ig : f, g is an element of A(X)}. We demonstrate that [A(X)](c) is a c-type subring of C(X, C) when A(X) is an element of R[X] is a c-type subring of C(X). Finally, for an intermediate subring A(X) of C(X), we show that the completion of X with respect to a suitable uniform structure derived from [A(X)]c is equal to VAX, the A-compactification of X.
. In this paper, we present generaized Sherman-Morrison-Woodbury formula for the g-Drazin inverse in a Banach algebra. New results for the g-Drazin invertibility of a modified element a - cdb with the generalized Schur complement dd - badc are given. The g-Drazin invertibility of an operator matrix with nonsingular generalized Schur complement under weaker restrictions is thereby established.
Recently, Zagreb energies, a graph invariant based on the eigenvalues of the Zagreb matrices have been proposed as an analogous to graph energy. In this communication, the Zagreb energies and Zagreb spectral radius are examined in relation to a number of graph operations, such as m-splitting graphs, m-shadow graphs, m-duplicate graphs, and extended bipartite double graphs. Further, we explore these generalised graphs within the context of specific graph types such as complete graphs, complete bipartite graphs, cycle graphs, and the complements of cycle graphs. Furthermore, we report an error present in [20] that contradicts the claim of hyperenergetic behaviour for the splitting graph of regular graphs, and also establish the non-hyperenergetic behaviour of the m-shadow graph.
The goal of this study is to provide an innovation for commutativity research that is less than the strong condition prime ring. This paper will describe weakly right cancellative semirings and examine how commutativity and generalized derivations apply to this class of semirings. A detailed explanation and classification of some of these generalized derivations are also included.
In (Categories and General Algebraic Structures with Applications, 12(1):175-197 (2020)), Rashidi et al. introduced GPW-flatness of acts over monoids as a generalization of principal weak flatness. In this paper, we introduce Condition (GPWP(sec)) of acts over monoids and compare it with GPW-flatness. Also, we obtain some general properties of Condition (GPWP(sec)) and characterize those monoids for which this condition implies some other properties and vice versa.
The goal of this study is to provide an innovation for commutativity research that is less than the strong condition prime ring. This paper will describe weakly right cancellative semirings and examine how commutativity and generalized derivations apply to this class of semirings. A detailed explanation and classification of some of these generalized derivations are also included.
For an ordered k-partition Omega( )= {S-1, S-2, ..., S-k} of vertex set of a connected graph C and a vertex v of C, the representation of v with respect to Omega ( )is defined as the k-tuple r(v|Omega) = (d(v, S-1), d(v, S-2), ..., d(v, S-k)). The partition S(2)is called a resolving partition of C, if r(u S-2) =/ r(v S-2) for all distinct u, v is an element of V(C). The partition dimension of a graph C, denoted by pd(C), is the cardinality of a minimum resolving partition of C. A subset D subset of V (C) is k-dominating in C, if every vertex of V(C) \ D has at least k neighbors in D. The minimum cardinality among all k-dominating sets is called the k-domination number of C, denoted by gamma(k)(C). In this paper, we determine the partition dimension of cocktail party graph CP(m+ 1) and corona product C degrees K-m(sic). Moreover, we obtain k-domination numbers for CP(m + 1) and corona product C-n degrees K-m(sic).
In this paper, we characterize the capability of nilpotent n-Lie algebras of dimension at most n + 3 over an arbitrary field when n > 2 and the capability of 7-dimensional nilpotent 3-Lie algebras over a field K with charK =/ 2.
. The property of having infinitely many prime numbers award these numbers to have many applications in various fields of sciences. One of the most important applications is their use in the creation of many public key cryptosystems' private keys. Therefore, the main aim of this paper is considering a well known form of primes generated by the Fermat-Euler equation p = x(2) + dy(2) and studying whether or not this form keeps the property of generating infinitely many primes if the unknowns x, y and p are terms in certain binary recurrence sequences called the Lucas sequences of the first kind {un(a, b)} or the second kind {vn(a, b)}. In other words, in this paper we present a technique for investigating the integer solutions (x, y, p) of the equation p = x(2 )+ dy(2), where the unknowns are terms in {un(a, b)} or {vn(a, b)}. We also apply this technique for determining the solutions (x, y, p) = (t(i)(a, b), tj(a, b), t(k)(a, b)) with 1 <= i <= j <= k, where t(n)(a, b) represents the general term u(n)(a, b) or v(n)(a, b) under certain conditions on the integers a and b.
In (Categories and General AlgebraicStructures with Applications, 12(1):175-197 (2020)), Rashidi et al. introduced GPW-flatnessof acts over monoids as a generalization of principalweak flatness.In this paper, we introduce Condition (GPWPsec) of acts overmonoids and compare it with GPW-flatness. Also, we obtainsome general properties of Condition (GPWPsec) and characterize those monoids for which this condition implies some otherproperties and vice versa.در( رسته ها و ساختارهای جبری عمومی با کاربردها 12(1):175-197(2020))، رشیدی و همکاران خاصیتهمواری - GPWسیستم ها روی تکواره ها را به عنوان تعمیمی از به طور اساسی ضعیف همواری معرفی کردند. در این مقاله شرط (〖GPWP〗_sec)از سیستم ها روی تکواره ها را ارائه و آن را با همواری - GPWمقایسه می کنیم. همچنین برخی از خواص کلی شرط (〖GPWP〗_sec). را به دست می آوریم و به دسته بندی تکوارهایی خواهیم پرداخت که این شرط سایر خواص را نتیجه دهد و برعکس
In this paper, we apply the concepts of intuitionistic fuzzy translations and multiplications of intuitionistic fuzzy sets to the PMS-subalgebras of a PMS algebra and investigate several related results. The relationships between intuitionistic fuzzy PMS-subalgebras of a PMS-algebra and intuitionistic fuzzy omega-translations, and intuitionistic fuzzy zeta-multiplications of the intuitionistic fuzzy PMS-subalgebras are discussed. Finally, we discuss the ideas of intuitionistic fuzzy magnified zeta omega translations of intuitionistic fuzzy PMS-subalgebras of a PMS-algebra and investigate associated results.
Let a be an ideal of a Noetherian ring R such that the R-modules H-a(1) (M) and H-a(3) (M) are a-cofinite, for all finitely generated R-modules M. In this paper, it is shown that the R-modules H-a(i)(M) are a-cofinite, for all finitely generated R-modules M and all integers i is an element of N-0.
