Journal of Algebraic Systems
https://jas.shahroodut.ac.ir/
Journal of Algebraic Systemsendaily1Sun, 01 Jan 2023 00:00:00 +0330Sun, 01 Jan 2023 00:00:00 +0330GENERALIZED FORMAL LOCAL COHOMOLOGY MODULES
https://jas.shahroodut.ac.ir/article_2953.html
Let $a$ be an ideal of a local ring $(R, m)$ and $M$ and &nbsp;$N$ two finitely generated &nbsp;&nbsp; $R$-modules. In this paper, we introduce the concept of generalized formal local cohomology modules. We define $i$-th generalized formal local cohomology module of $M$ and $N$ with respect to &nbsp;&nbsp;&nbsp; $a$ by &nbsp;$\mathfrak{F}_{a}^i(M,N) := \underset{n}{\varprojlim}H_m^i(M,N/{a}^{n}N )$ for $i\geq 0$. We prove several results concerning vanishing and finiteness properties of these modules.WEAKLY PRIME AND SUPER-MAX FILTERS IN BL-ALGEBRAS
https://jas.shahroodut.ac.ir/article_2954.html
&lrm;In this paper&lrm;, &lrm;the concepts of weakly prime filters and super-max filters in $\mathrm{BL}$-algebras are introduced&lrm;, &lrm;and the relationships between them are discussed&lrm;. &lrm;Also&lrm;, &lrm;some properties and relations between these filters and other types of filters in $\mathrm{BL}$-algebras are given&lrm;. &lrm;With some examples&lrm;, &lrm;it is shown that these filters have differences&lrm;. &lrm;After that&lrm;, &lrm;the notions of weakly linear $\mathrm{BL}$-algebras and weak top $\mathrm{BL}$-algebras are defined and investigated&lrm;. &lrm;Finally&lrm;, &lrm;using the notion of a weakly prime filter&lrm;, &lrm;a new topology on $\mathrm{BL}$-algebras is defined and studied&lrm;.A GRAPH ASSOCIATED TO ESPECIAL ESSENTIALITY OF SUBMODULES
https://jas.shahroodut.ac.ir/article_2956.html
&lrm;Let $R$ be an associative ring with identity&lrm;. &lrm;In this paper we&lrm;&lrm;associate to every $R$-module $M$ a simple graph $\Gamma_e(M)$&lrm;, which we call it the essentiality graph of $M$. The vertices of $\Gamma_e(M)$ are nonzero submodules of $M$ and two distinct&lrm;&lrm;vertices $K$ and $L$ are considered to be adjacent if and only&lrm;&lrm;if $K\cap L$ is an essential submodule of $K+L$&lrm;.&lrm;&lrm;We investigate the relationship between some module theoretic&lrm;&lrm;properties of $M$ such as minimality and closedness of&lrm;&lrm;submodules with some graph theoretic properties of&lrm;&lrm;$\Gamma_e(M)$&lrm;. &lrm;In general&lrm;, &lrm;this graph is not connected&lrm;. &lrm;We&lrm;&lrm;study some special cases in which $\Gamma_e(M)$ is&lrm;&lrm;complete or a union of complete connected components and give some examples illustrating each specific case&lrm;.SOME RESULTS ON ORDERED AND UNORDERED FACTORIZATION OF A POSITIVE INTEGER
https://jas.shahroodut.ac.ir/article_2957.html
A well-known enumerative problem is to count the number of ways a positive integer $n$ can be factorised as $n=n_1\times n_2\times\cdots\times n_{k}$, where $n_1\geqslant n_2 \geqslant \cdots \geqslant n_{k} &gt;1$. In this paper, we give some recursive formulas for the number of ordered/unordered factorizations of a positiveinteger $n$ such that each factor is at least $\ell$. In particular, by using elementary techniques, we give an explicit formula in cases where $k=2,3,4$.ON CLOSED HOMOTYPICAL VARIETIES OF SEMIGROUPS-2
https://jas.shahroodut.ac.ir/article_2958.html
In this paper we extended the results of paper\linebreak ``On Closed Homotypical Varieties of Semigroups" and have shown that the homotypical varieties of semigroups defined by the identities &nbsp;$axy=x^nayx$, $axy=xa^nya$[$axy=yay^nx$],$axy=xaya^n$[$axy=y^nayx$] and $axy=xayx^n$ are closed in itself, where $(n \in \mathbb{N})$.REGULAR FILTERS OF DISTRIBUTIVE LATTICES
https://jas.shahroodut.ac.ir/article_2959.html
The concepts of regular filters and &pi;--filters are introduced in distributive lattices. A set of equivalent conditions is given for a D-filter to become a regular filter. For every D-filter, it is proved that there exists a homomorphism whose dense kernel is a regular filter. &pi;--filters are characterized in terms of regular filters and congruences. Some equivalent conditions are given for the space of all prime &pi;-filters to become a Hausdorff space.COMMON NEIGHBOURHOOD SPECTRUM AND ENERGY OF COMMUTING CONJUGACY CLASS GRAPH
https://jas.shahroodut.ac.ir/article_2960.html
In this paper, we compute the common neighbourhood (abbreviated as CN) spectrum and the common neighbourhood energy of commuting conjugacy class graph of several families of finite non-abelian groups. As a consequence of our results, we show that the commuting conjugacy class graphs of the groups $D_{2n}$, $T_{4n}$, $SD_{8n}$, $U_{(n,m)}$, $U_{6n}$, $V_{8n}$, $G(p, m, n)$ and some families of groups whose central quotient is isomorphic to $D_{2n}$ or $\mathbb{Z}_p \times \mathbb{Z}_p$, for some prime $p$, are CN-integral but not CN-hyperenergetic.ADMISSIBLE (REES) EXACT SEQUENCES AND FLAT ACTS
https://jas.shahroodut.ac.ir/article_2961.html
Let $S$ be a commutative pointed monoid. In this paper, some properties of admissible (Rees) short exact sequences of $S$-acts are investigated. In particular, it is shown that every admissible short exact sequence of $S$-acts is Rees short exact. In addition, a characterization of flat acts via preserving admissible short exact sequences is established. As a consequence, we show that for a flat $S$-act $F$, the functor $F \otimes_{S} -$ preserves admissible morphisms. Finally, it is proved that the class of flat $S$-acts is a subclass of admissibly projective ones.Generalized π-Baer *-rings
https://jas.shahroodut.ac.ir/article_2962.html
&lrm;A *-ring&lrm; &lrm;$R$ is called a generalized $\pi$-Baer *-ring&lrm;, &lrm;if for any projection invariant left ideal $Y$ of $R$&lrm;, &lrm;the right annihilator of $Y^n$&lrm; &lrm;is generated&lrm;, &lrm;as a right ideal&lrm;, &lrm;by a projection, &lrm;for some positive integer $n$&lrm;, &lrm;depending on $Y$&lrm;. &lrm;In this paper, &lrm;we&lrm; &lrm;study some properties of generalized $\pi$-Baer *-rings&lrm;. &lrm;We show that this notion is well-behaved with respect to&lrm; &lrm;polynomial extensions, full matrix rings, and several classes of triangular matrix rings&lrm;.&lrm; &lrm;We indicate interrelationships between the generalized $\pi$-Baer *-rings and related classes of rings such as&lrm; generalized &lrm;$\pi$-Baer rings&lrm;, generalized &lrm;Baer *-rings&lrm;, generalized quasi-Baer *-rings, and &lrm;$&lrm;\pi$-Baer \s-rings. &lrm;We obtain algebraic examples which are generalized&lrm; $&lrm;\pi$-Baer $ \ast $-rings but are not $&lrm;\pi$-Baer *-rings&lrm;. &lrm;We show that for pre-C*-algebras these two notions are equivalent&lrm;.&lrm;We obtain classes of Banach *-algebras&lrm; &lrm;which are generalized&lrm; $&lrm;\pi$-Baer *-rings but are not $&lrm;\pi$-Baer *-rings&lrm;. We finish the paper by showing that for a locally compact&lrm;&lrm;abelian group $G$&lrm;, &lrm;the group algebra $L^{1}(G)$ is a&lrm; &lrm;generalized $&lrm;\pi$-Baer $*$-ring&lrm;, &lrm;if and only if so is the group C*-algebra&lrm; &lrm;$C^{*}(G)$&lrm;, &lrm;if and only if $G$ is finite&lrm;.AN IDENTITY RELATED TO θ-CENTRALIZERS IN SEMIPRIME RINGS
https://jas.shahroodut.ac.ir/article_2963.html
&lrm;Let $R$ be a $ 2$-torsion-free semiprime ring and $\theta$ be an epimorphism of $R$&lrm;. &lrm;In this paper&lrm;, &lrm;under special hypotheses&lrm;, we prove that if $T&lrm;: &lrm;R\longrightarrow R$&lrm; &lrm;is an additive mapping such that&lrm;&lrm;$&lrm;&lrm;$&lrm;&lrm;T(xyx)=&theta;(x)T(y)&theta;(x)&lrm;,&lrm;$&lrm;&lrm;$&lrm;&lrm;holds for all $x&lrm;, &lrm;y\in R$&lrm;, &lrm;then&lrm; &lrm;$T$ is a $&theta;$-centralizer&lrm;either $R$ is unital&lrm; or $&theta;(Z(R))=Z(R)$.JACOBSON MONOFORM MODULES
https://jas.shahroodut.ac.ir/article_2966.html
In this paper, we introduce and study the concept of Jacobson monoform modules whichis a proper generalization of that of monoform modules. We present a characterization of semisimplerings in terms of Jacobson monoform modules by proving that a ring $R$is semisimple if and only if every $R$-module is Jacobson monoform. Moreover, we demonstrate that over a ring $R$, the properties monoform, Jacobson monoform, compressible, uniform and weakly co-Hopfian are all equivalent.FURTHER STUDIES OF THE PERPENDICULAR GRAPHS OF MODULES
https://jas.shahroodut.ac.ir/article_2967.html
&lrm;In this paper we continue our study of perpendicular graph of modules&lrm;, &lrm;that was introduced in \cite{Hokkaido}&lrm;. &lrm;Let $R$ be a ring and $M$ be an $R$-module&lrm;. &lrm;Two modules $A$ and&lrm; &lrm;$B$ are called orthogonal&lrm;, &lrm;written $A\perp B$&lrm;, &lrm;if they do not have&lrm; &lrm;non-zero isomorphic submodules&lrm;. &lrm;We associate a graph $\Gamma_{\bot}(M)$ to $M$&lrm; &lrm;with vertices&lrm; &lrm;$\mathcal{M}_{\perp}=\{(0)\neq A\leq M\;|\; \exists (0)\neq B\leq M \; \mbox{such that}\; A\perp B\}$&lrm;, &lrm;and for distinct $A,B\in&lrm; &lrm;\mathcal{M}_{\perp}$&lrm;, &lrm;the vertices $A$ and $B$ are adjacent if and only if&lrm; &lrm;$A\perp B$&lrm;. &lrm;The main object of this article is to study the&lrm; &lrm;interplay of module-theoretic properties of $M$ with&lrm; &lrm;graph-theoretic properties of $\Gamma_{\bot}(M)$&lrm;. &lrm;We study the clique number and chromatic number of $\Gamma_{\bot}( M)$&lrm;. &lrm;We prove that if $\omega(\Gamma_{\bot}( M)) &lt; \infty $ and $M$ has a simple submodule&lrm;, &lrm;then $\chi(\Gamma_{\bot}(M)) &lt; \infty $&lrm;. &lrm;Among other results&lrm;, &lrm;it is shown that for a semi-simple module $M$&lrm;, &lrm;$\omega(\Gamma_{\bot}(_R M))=\chi(\Gamma_{\bot}(_R M))$&lrm;.(f, g)-DERIVATIONS IN RESIDUATED LATTICE
https://jas.shahroodut.ac.ir/article_3095.html
In this paper, we present and examine the characteristics of (f, g)-derivations for a residuated lattice. Some relationships between (f, g)-derivationand isotone, contractive and ideal (f, g)-derivations are given. The set of fixedpoint of an (f, g)-derivation is introduced and its structure is studied. More precisely, we show that the set of fixed points is also a residuated lattice.On the finiteness of local homology modules
https://jas.shahroodut.ac.ir/article_3096.html
Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ be an ideal of $R$. Suppose $M$ is a finitely generated $R$-module and $N$ is an Artinian $R$-module. We define the concept of filter coregular sequence to determine the infimum of integers $i$ such that the generalized local homology $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is not finitely generated as an $\widehat{R}^{\mathfrak{a}}$-module, where $\widehat{R}^{\mathfrak{a}}$ denotes the $\mathfrak{a}$-adic completion of $R$. In particular, if $R$ is a complete semi-local ring, then $\textrm{H}^{\mathfrak{a}}_i(M, N)$ is a finitely generated $\widehat{R}^{\mathfrak{a}}$-module for all non-negative integers $i$ if and only if $(0:_N\mathfrak{a}+\textrm{Ann}(M))$ has finite length.Some algebraic and measure theoretic properties of the rings of measurable functions
https://jas.shahroodut.ac.ir/article_3097.html
Let $M(X, \mathcal{A}, \mu)$ be the ring of real-valued measurable functionson a measure space $(X, \mathcal{A}, \mu)$. In this paper, we show that the maximal ideals of $M(X, \mathcal{A}, \mu)$ are associated with the special measurable sets in $\mathcal{A}$. We also study some other algebraic properties of $M(X, \mathcal{A}, \mu)$.ON THE COMINIMAXNESS OF LOCAL COHOMOLOGY MODULES
https://jas.shahroodut.ac.ir/article_3098.html
Let I be an ideal of a commutative Noetherian ring R. It is shown thatthe R-modules HiI (M) are I-cominimax, for all finitely generated R-modules M and alli &isin; N0, if the R-modules HiI (R) are I-cofinite with dimension not exceeding 1, for allintegers i &ge; 2. This is an analogue result of Bahmanpour in [11].Domination Number and Identifying Code Number of the Subdivision Graphs
https://jas.shahroodut.ac.ir/article_3099.html
&lrm;Let $G=(V&lrm;, &lrm;E)$ be a simple graph&lrm;. &lrm;A set $C$ of vertices of $G$ is an identifying code of $G$ if for every two vertices $x$ and $y$ the sets $N_{G}[x] \cap C$ and $N_{G}[y] \cap C$ are non-empty and different&lrm;. &lrm;Given a graph $G,$ the smallest size of an identifying code of $G$ is called the identifying code number of $G$ and denoted by $\gamma^{ID}(G).$ In this paper&lrm;, &lrm;we prove that the identifying code number of the subdivision of a graph $G$ of order $n$ is at most $n$&lrm;. &lrm;Also&lrm;, &lrm;we prove that the identifying code number of the subdivision of graphs $K_n$, $K_{r,s}$ and $C_P(s)$ are $n&lrm;$,&lrm; &lrm;&lrm;&lrm;$&lrm;&lrm;r+s$ and $2s$, respectively&lrm;. &lrm;Finally&lrm;, &lrm;we conjecture that for every graph $G$ of order $n$ the identifying code number of the subdivision of $G$ is $n$&lrm;.FIXED POINTS AND CUT-HOMOMORPHISMS GENERATED BY ACTIONS OF A BE-ALGEBRA ON ITS SUBALGEBRA
https://jas.shahroodut.ac.ir/article_3100.html
The concept of actions of a BE-algebra on its subalgebra is introduced and certain properties of these actions are derived. The notion of cut-homomorphisms is introduced in BE-algebras and proved that the class of all cut-homomorphisms forms an ordered BE-algebra. Properties of fixed points of cut-homomorphisms of BE-algebras are investigated and a set of equivalent conditions is given for any twocut-homomorphisms are equal in the sense of mappings.On a skew generalized triangular matrix ring
https://jas.shahroodut.ac.ir/article_3101.html
In this article, we study skew monoid rings in which the monoid used in their structure is a quotient of a free monoid. We study annihilator conditions of them and describe conditions for transferring some properties from the base ring $R$ to these extensions. Interesting examples are provided for properties that are not transferred from $R$ to these extensions.Some operator inequalities in Hilbert $C^{*}$-modules via the operator perspective
https://jas.shahroodut.ac.ir/article_3102.html
Some Hilbert $C^*$-module versions of H$\ddot{o}$lder-McCarthy and H$\ddot{o}$lder type inequalities and their complementary on a Hilbert $C^*$-module are obtained by Seo \cite{seo-2014}. The purpose of this paper is to extend these results for some operator convex (resp. concave) functions on a Hilbert $C^*$-module via the operator perspective approach. By choosing some elementary functions, we reach some new types of inequalities in Hilbert $C^*$-modules.Fault-Tolerant Metric Dimension of Annihilator Graphs of Commutative Rings
https://jas.shahroodut.ac.ir/article_3103.html
Let R be a commutative ring with identity. The annihilator graph AG (R) is a simple graph with vertex set as the set of all non-zero zero-divisors of R, and two distinct vertices a and b are adjacent if and only if annR (a) &cup; annR (b) ̸= annR (a &middot; b). We depicted the relationship between the fault-tolerant metric dimension of AG (R) and some graph parameters. Furthermore, we computed the fault-tolerant metric dimension of the annihilator graph of reduced and non-reduced rings.Applications of Rough Soft to Extensions Semihypergroups Induced by Operators and Corresponding Decision Making Methods
https://jas.shahroodut.ac.ir/article_3104.html
In this paper, we apply a rough soft set to a spe-cial algebraic hyperstructure, and give the concept of a rough soft semihypergroup. We propose the notion of lower and upper approximations concerning a special semihypergroup and obtain some properties. Moreover, we consider a connection between the lower(upper) approximation of a special semihypergroup and the lower(upper) approximation of the associated -hypergroupoid. In the last section of this research, we discuss the decision-making algorithm of rough soft semihypergroups. Afterward, we obtain a relation between the decision-making algorithm of rough soft semi-hypergroups and their associated rough soft -hypergroupoids for a special semihypergroup.HEMI-COMPLEMENTED LATTICES
https://jas.shahroodut.ac.ir/article_3105.html
The notion of hemicomplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemicomplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemicomplemented lattice to become a D-Stone lattice. Hemicom-complemented lattices and D-Stone lattices are characterized in topological terms.A NOTE ON NONLINEAR MIXED ∗-JORDAN TYPE DERIVATIONS ON ∗-ALGEBRAS
https://jas.shahroodut.ac.ir/article_3106.html
Let S be a &lowast;-algebra containing the unity and a nontrivial projection. In the present paper, we showthat under certain restrictions if a map &Psi; : S &rarr; S satisfies &Psi;(L ⋄ N &bull; D) = &Psi;(L) ⋄ N &bull; D + L ⋄&Psi;(N) &bull; D + L ⋄ N &bull; &Psi;(D) for all L, N, D &isin; S, then &Psi; is an additive &lowast;-derivation.Equitable Rings Domination in Graphs
https://jas.shahroodut.ac.ir/article_3107.html
A dominating set $S$ of $G$ is an \textit{equitable dominating set} of $G$ if for every $v \in V(G) \setminus S$, there exists $u \in S$ such that $uv \in V(G)$ and $\displaystyle{\left|\deg(u) - \deg(v)\right| \leq 1.}$ A dominating set $S$ of $G$ is a \textit{rings dominating set} of $G$ if every vertex $v \in V(G) \setminus S$ is adjacent to atleast two vertices $V(G) \setminus S$. In this paper, we examine the conditions at which the equitable dominating set and the rings dominating set coincide, and thus naming the dominating set as \textit{equitable rings dominating set}. The minimum cardinality of an equitable rings dominating set of a graph $G$ is called the \textit{equitable rings domination number} of $G$, and is denoted by $\gamma_{eri}(G)$. Moreover, we examine determine the equitable rings domination number of many graphs, and graphs formed by some binary operations.A study on Tri reversible Rings
https://jas.shahroodut.ac.ir/article_3108.html
This article embodies a ring theoretic property which,preserves the reversibility of elements at non-zero tripotents. Aring R is defined as quasi tri reversible if any non-zero tripotentelement ab of R implies ba is also a tripotent element in R fora, b &isin; R. We explore the quasi tri reversibility of 2 by 2 full andupper triangular matrix rings over various kinds of reversible rings,deducing that the quasi tri reversibility is a proper generalizationof reversible rings. It is proved that the polynomial rings are notquasi tri reversible rings. The relation of symmetric rings, IF Pand Abelian rings with reversibility and quasi tri reversibility arestudied. It is also observed that the structure of weakly tri normalrings and quasi tri reversible rings are independent of each other.Some identities involving endomorphisms of prime rings
https://jas.shahroodut.ac.ir/article_3109.html
In this paper we will extend some results on the commutativity of quotient rings proved in [1] and [11]. However, we will consider endomorphisms instead of derivations and generalized derivations, which is sufficient to obtain good results. We will also show that the conditions imposed in that paper cannot be removed.w−filters of Almost Distributive Lattices
https://jas.shahroodut.ac.ir/article_3110.html
The notion of w&minus;filters is introduced in an Almost Distributive Lattice(ADL) and properties are investigated. A necessary and sufficient condition is derived for a maximal filter of anADL to become a w&minus;filter which leads to a characterization of a quasi-complemented ADL. Also, w&minus;filters of an ADL are characterized in terms of minimal prime D&minus;filters.A SURVEY ON χ-MODULE CONNES AMENABILITY OF SEMIGROUP ALGEBRAS
https://jas.shahroodut.ac.ir/article_3111.html
We shall study the &chi;-module Connes amenability of a semigroup algebra l^1(S),where &chi; is a bounded module homomorphism from l^1(S) to l^1(S) that is w^&lowast;-continuous and S is an inverse weakly cancellative semigroup with subsemigroup E of idempotents. We are mainly concerned with the study of &chi;-module normal, virtual diagonals. We characterize the &chi;- module Connes amenability of a semigroup algebra l^1(S). Also, we show that if l^1(S) as a Banach module over l^1(E) has an id-module normal, virtual diagonal then it is id-module Connes amenable. Other characterizations of &chi;- module Connes amenability of l^1(S) is presented.Fuzzy Neutrosophic Prime Ideals of BCK-algebras.
https://jas.shahroodut.ac.ir/article_3112.html
In this research paper, we introduce and analyze the notion of fuzzy neutrosophic prime ideals (FNPIs) in a commutative BCK-algebra $\mathcal{K}$. It represents a further extension of prime ideals in the context of fuzzy neutrosophic sets. We provide an example that shows that not every fuzzy neutrosophic ideal of a commutative BCK-algebra $\mathcal{K}$ is a FNPI of $\mathcal{K}$. We also prove that a fuzzy neutrosophic set of $\mathcal{K}$ is a FNPI of $\mathcal{K}$ if, for all $a,b,c \in [0,1] $, the upper (a,b)-level cut and lower c-level cut are prime ideals of $\mathcal{K}$.On $(m,n)$-ary $P$-$H_{v}$-modules and their isomorphism theorems
https://jas.shahroodut.ac.ir/article_3113.html
After introducing the definition of hypergroups by Marty, the study of hyperstructures and its generalization to $(m,n)$-ary hyperstructures has been of great importance. In this paper, we construct the structure of $(m,n)$-ary $H_v$-modules over $(m,n)$-ary $H_v$-rings by using the notion of $P$-hyperoperations. We study their properties and prove their isomorphism theorems.A NON-COMMUTATIVE GENERALIZATION OF MTL-RINGS
https://jas.shahroodut.ac.ir/article_3114.html
The current work extends the class of commutative MTL-rings established by the authors to the non-commutative ones. That class of rings will be named generalized MTL-rings since they are not necessary commutative. We show that in the non-commutative case, a local ring with identity is a generalized MTL-ring if and only if it is an ideal chain ring. We prove that the ring of matrices over an MTL-ring is a non-commutative MTL-ring. We also study their representation in terms of subdirectirreducibility.FUZZY PERSPECTIVITY IN FUZZY LATTICES
https://jas.shahroodut.ac.ir/article_3115.html
In this paper, we have introduced and studied the notion of perspectivity in fuzzy lattices. The motivation is from the work done by Wasadikar and Khubchandani. We have tried to relate $\bigtriangledown_F$-relation with fuzzy perspective relation. Also, we prove that for a pair of a fuzzy atoms, the concept of fuzzy subperspective holds. Subsequently, several related properties are proven.On $\mathcal{Z}$-symmetric modules
https://jas.shahroodut.ac.ir/article_3116.html
A ring $R$ is called a left $\mathcal{Z}$-symmetric ring if $ab \in \mathcal{Z}_l(R)$ implies $ba \in \mathcal{Z}_l(R)$, where $\mathcal{Z}_l(R)$ is the set of left zero-divisors. A right $\mathcal{Z}$-symmetric ring is defined similarly, and a $\mathcal{Z}$-symmetric ring is one that is both left and right $\mathcal{Z}$-symmetric. In this paper, we introduce the concept of $\mathcal{Z}$-symmetric modules as a generalization of $\mathcal{Z}$-symmetric ring. Additionally, we introduce the concept of an eversible module as an analogy to eversible rings and prove that every eversible module is also a $\mathcal{Z}$-symmetric module, like in the case of rings.More on total domination polynomial and $\mathcal{D}_t$-equivalence classes of some graphs
https://jas.shahroodut.ac.ir/article_3117.html
Let $G = (V, E)$ be a simple graph of order $n$. A total dominating set of $G$ is a subset $D$ of $V$ such that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to the minimum cardinality of a total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of all graphs $\mathcal{D}_t$-equivalent to $G$. A polynomial $\sum_{k=0}^n a_kx^k$ is called unimodal if the sequence of its coefficients is unimodal, that means there is some $k \in \{0, 1, \ldots , n\}$, such that $a_0 \leq \ldots \leq a_{k-1} \leq a_k\geq a_{k+1} \geq \ldots \geq a_n$. In this paper, we investigate $\mathcal{D}_t$-equivalence classes of some graphs. Also, we introduce some families of graphs whose total domination polynomials are unimodal.The $\mathcal{D}_t$-equivalence classes of graphs of order $\leq 6$ are presented in the appendix.On weak extended order algebras with adjoint pairs and Galois pairs
https://jas.shahroodut.ac.ir/article_3118.html
In this paper, we consider properties of weak extended order algebras with adjoint pairs and Galois pairs, and prove some new results. Moreover, we clarify the relation between these algebras and BCK-algebras, that is, the class of all normal weak extended order algebras with adjoint pair satisfying the condition $\top \to x=x$ is identical the class of all BCK-algebras with the condition (S).Weak Idempotent Nil-clean Rings
https://jas.shahroodut.ac.ir/article_3119.html
We introduce the concept of a weak idempotent nil-clean ring as a generalization of a weakly nil-clean ring. We give certain characterizations for weak idempotent nil-clean rings in terms of Jacobson radical and nil radical. Further, we obtain any weak idempotent nil-clean ring is a direct product nil clean rings in terms of Jacobson radicals .On the nilpotent dot product graph of a commutative ring
https://jas.shahroodut.ac.ir/article_3120.html
Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m&lt;\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) graph, total nilpotent dot product graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in $\mathcal{N(R)}$, \mbox{for some }z\in \mathcal{R}^*\} $. The two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in $\mathcal{N(B)}$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N(B)}$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS
https://jas.shahroodut.ac.ir/article_3121.html
An ideal $I$ of a ring $R$ is called a right strongly Baer ideal if $r(I)=r(e)$, where $e$ is an idempotent, and there are right semicentral idempotents $e_{i}$ ($1\leq i\leq n$) with $ReR=Re_{1}R\cap Re_{2}R\cap...\cap Re_{n}R$ and each ideal $Re_{i}R$ is maximal or equals $R$. In this paper, we provide a topological characterization of this class of ideals in semiprime (resp., semiprimitive) rings. By using these results, we prove that every ideal of a ring $R$ is a right strongly Baer ideal \textit{if and only if} $R$ is a semisimple ring. Next, we give a characterization of right strongly Baer-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, and semiprime rings. For a semiprimitive commutative ring $R$, it is shown that $\Soc(R)$ is a right strongly Baer ideal \textit{if and only if} the set of isolated points of $\Max(R)$ is dense in it \textit{if and only if} $\Soc_{m}(R)$ is a right strongly Baer ideal. Finally, we characterize strongly Baer ideals in $C(X)$ (resp., $C(X)_{F}$).On the diameter and connectivity of bipartite Kneser type-k graphs
https://jas.shahroodut.ac.ir/article_3122.html
Let $n\in \mathbb{Z}^{+}$, $n&gt;1$, and $k$ be an integer, $1\leq k \leq n-1$. The graph $H_{T}(n,k)$ is defined as a graph with vertex set $V$, as all non-empty subsets of $\mathcal{S}_n=\{1,2,3,\ldots,n\}$. It is a bipartite graph with partition $(V_{1},V_{2})$, in which $V_{1}$ contains the $k$-element subsets of $\mathcal{S}_n$ and $V_{2}$ contains $i$-element subsets of $\mathcal{S}_n$, for $1\le i \le n, \ i\neq k$. An edge exists between a vertex $U\in V_{1}$ and a vertex $W \in V_{2}$ if either $U\subset W$ or $W \subset U$. In this paper, we established formulae for the number of edges, vertex connectivity, edge connectivity and degree polynomial. Then, we analysed the clique number and diameter of $H_T(n,k)$. We also verified that the graphs $H_{T}(n,k)$ and $H_{T}(n,n-k)$ are isomorphic. Degree-based topological indices such as the general Randic connectivity index, the first general Zagreb index and the general sum connectivity index are also computed. Also, we proved that the line graph of $H_T(n,k)$ is not a bipartite graph.Zero Forcing Number and Maximum Nullity of General Power Graphs
https://jas.shahroodut.ac.ir/article_3123.html
Let &Gamma; = (V,E) be a simple and undirected graph. General power graph of &Gamma;, shown by Pg(&Gamma;), is a graph with the vertex set P(V (&Gamma;))\ϕ. Also two distinct vertices of B and C are adjacent if and only if every b &isin; B is adjacent to every c &isin; C \{b} in &Gamma;. In this paper, we consider general power graph related to graph &Gamma;. Also we show that zero forcing number is equal to maximum nullity, for general power graph of some graphs.K-FILTERS OF DISTRIBUTIVE LATTICES
https://jas.shahroodut.ac.ir/article_3124.html
The concept of K-filters is introduced in distributive lattices and studied some properties of these classes of filters. Some necessary and sufficient conditions are derived for every &pi;-filter of a distributive lattice to become a K-filter. Some equivalent conditions are derived for every D-filter of a distributive lattice to become a K-filter. Quasi-complemented lattices are characterized with the help of K-filters.True-False structures and its application in groups and BCK/BCI-algebras
https://jas.shahroodut.ac.ir/article_3125.html
By utilizing the concept of interval-valued fuzzy sets, a novel structure called True-False structures is introduced. Its application in groups and BCK/BCI-algebras is discussed. The introduction of (limited) T &amp;F -subgroups and (limited) T &amp;F -subalgebras is carried out, along with an investigation of various associated properties. Characterizations of (limited) T&amp;F- subgroups and (limited) T&amp;F-subalgebras are provided, and it is demonstrated that the in- tersection of two T &amp;F -subgroups (respectively, T &amp;F -subalgebras) also forms a T &amp;F -subgroup (respectively, T &amp;F -subalgebra). Additionally, the union and intersection of two T &amp;F -subgroups (respectively, T&amp;F-subalgebras) are explored.Total near - ring graph
https://jas.shahroodut.ac.ir/article_3126.html
Let N be a right near-ring. Let Z(N) be the set of right zero-divisors of N.We dene total near-ring graph of N as a graph whose vertex set is the set of allelements of the near-ring N and any two distinct vertices n1; n2 2 N are adjacentif and only if n1 + n2 or n2 + n1 2 Z(N). We denote total near-ring graph of Nby TN. In this paper we try to give an overview of the structure of TN dependingupon whether the set of right zero-divisors Z(N) is an ideal of N or not. We alsond the girth and diameter of TN and its two subgraphs TZ(N) and TReg(N) for thecase when Z(N) is an ideal and not an ideal of the near-ring N.Genus of commuting graphs of certain finite groups
https://jas.shahroodut.ac.ir/article_3127.html
The commuting graph of a finite group G is a graph whose vertex set is the set of non-central elements of G and two distinct vertices are adjacent if they commute. In this article, we compute genus of commuting graphs of certain classes of finite non-abelian groups and characterize those groups such that their commuting graphs have genus 4, 5 and 6.Structured condition pseudospectra and structured essential condition pseudospectra of bounded linear operators on ultrametric Banach spaces
https://jas.shahroodut.ac.ir/article_3128.html
In this paper, we introduce and study the structured condition pseudospectra and the structured essential condition pseudospectra of bounded linear operators on ultrametric Banach spaces. We establish a characterization of the structured condition pseudospectrum of continuous linear operators and we give a relationship between the structured condition pseudospectrum and the structured pseudospectrum of continuous linear operators on ultrametric Banach spaces. Many characterizations of the structured essential condition pseudospectrum of bounded linear operators and examples are given.Results on quotient near-rings involving additive maps
https://jas.shahroodut.ac.ir/article_3129.html
We consider N to be a 3-prime field and P to be a prime ideal of N : In this paper, we studythe commutativity of the quotient ring N =P with left multipliers and derivations satisfying certainidentities on P, generalizing some well-known results in the literature. Furthermore, an example isused to illustrate the necessity of our hypothesesJNB-algebras
https://jas.shahroodut.ac.ir/article_3130.html
As a generalization of the self-distributive BE-algebra, the JNB-algebra is introduced, and its basic properties are investigated. This could play various roles in the study of logical algebra, including BCK-algebra. First, examples are presented showing that the three axioms of JNB-algebra are independent of each other. The basic properties of JNB-algebras that will be needed to study various theories about JNB-algebras are explored. Upper sets based on one and two elements are introduced and its associated properties are examined. Two concepts so called JNB-deductive system and JNB-filter are introduced, and their properties are investigated. Characterizations of the JNB-deductive system and the JNB-filter are discussed. It is finally confirmed that the JNB-deductive system matches the JNB-filter.Dedekind-MacNeille Completion of the Rough Sets System as Pasting of Rough Approximation Lattices
https://jas.shahroodut.ac.ir/article_3131.html
The pattern of embedding of rough approximation lattices defined by a reflexive relation in its Dedekind-MacNeille completion of rough sets system is taken up for study in this work.The reflexive relation R for which the Dedekind-MacNeille completion of rough sets system defined by R is the pasting of its rough approximation lattices is characterized. Some properties of theDedekind-MacNeille completion of rough sets system defined by areflexive relation R are also discussed.A new class of small submodules
https://jas.shahroodut.ac.ir/article_3132.html
&lrm;Let $R$ be a commutative ring with identity &lrm;$&lrm;1\neq 0&lrm;$ &lrm;and $M$ a nonzero unital $R$-module&lrm;. &lrm;In this paper&lrm;, &lrm;we introduce a new notion of submodules in $M$&lrm;, &lrm;namely $T$-semi-annihilator small submodules of $M$ with respect to an arbitrary submodule $T$ of $M&lrm;$&lrm;&lrm;. &lrm;A submodule $N$ of $M$ is $T$-semi-annihilator small in $M$ provide that for each submodule $X$ of $M$ with&lrm; &lrm;$T\subseteq N+X$ implies that ${\rm Ann}(X)\ll (T:M)$&lrm;. &lrm;In addition&lrm;, &lrm;we investigate some results concerning to this new class of submodules&lrm;. &lrm;Among various results&lrm;, &lrm;we prove that for a faithful finitely generated multiplication module $M$&lrm;, &lrm;the submodule $N$ of $M$ is a $T$-semi-annihilator small submodule of $M$ if and only if $(N:M)$ is a $(T:M)$-semi-annihilator small ideal of $R$&lrm;. &lrm;Finally&lrm;, &lrm;we explore the properties and the behaviour of this structure under ring homomorphism&lrm;, &lrm;localization&lrm;, &lrm;direct sums and tensor product of them with a &lrm;faithfully&lrm; flat $R$-module&lrm;.(weakly) $(s,n)$-closed hyperideals in commutative multiplicative hyperrings
https://jas.shahroodut.ac.ir/article_3133.html
&lrm;A multiplicative hyperring is a well-known type of algebraic hyperstructures which extends a ring to a structure in which the addition is an operation but the multiplication is a hyperoperation&lrm;. &lrm;Let $G$ be a commutative multiplicative hyperring and $s,n \in \mathbb{Z}^+$&lrm;. &lrm;A proper hyperideal $Q$ of $G$ is called (weakly) $(s,n)$-closed if ($0 \notin a^s \subseteq Q$ ) $a^s \subseteq Q$ for $a \in G$ implies $a^n \subseteq Q$&lrm;. &lrm;In this paper&lrm;, &lrm;we aim to investigate (weakly) $(s,n)$-closed hyperideals and give some results explaining the structures of these notions&lrm;.On type Krull dimension of modules
https://jas.shahroodut.ac.ir/article_3134.html
In this paper, the concept of type Krull dimension of a module is introduced and some related properties are investigated.Using this concept, we extend some basic results about modules with Krull dimension. It is shown that every module with homogeneous type Krull dimension has type Krull dimension equal to zero. Also, it is proved that an $R$-module $M$ has type Krull dimension if and only if it has type Noetherian dimension. We observe that, every module with Krull dimension has type Krull dimension, but its converse is not true in general. Further, we define t-Artinian (resp., t-Noetherian) modules and it is shown that if $M$ be a t-Artinian $R$-module with type Krull dimension, then it has Krull dimension and these two dimensions for $M$ coincide. At the end, we define the concept of $\alpha$-DICCT modules and it is proved that an $R$-module $M$ is $\alpha$-DICCT if and only if it has type Krull dimension.SHEFFER STROKE BE-ALGEBRAS BASED ON THE SOFT SET ENVIRONMENT
https://jas.shahroodut.ac.ir/article_3135.html
In this paper, we handle the concept of Sheffer stroke BE-algebras within the framework of soft sets. We introduce the notion of Sheffer stroke BE-algebras based on the soft set environment, providing a novel perspective on their algebraic properties. These soft Sheffer stroke BE-algebras extend a flexible and adaptable approach to logical operations, allowing for the combination of fuzzy and crisp information. Furthermore, we reveal the concept of soft Sheffer stroke sub-BE-algebras and we investigate the properties of these structures. Our analysis put forward intriguing connections among soft sets, Sheffer stroke operations, and the underlying BE-algebraic structure. The results given in this paper contribute to the broader understanding of algebraic structures basis with Sheffer stroke operation in the context of soft sets and provide potential applications in fields such as decision-making, information fusion, and uncertain reasoning.