Shahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901THE ANNIHILATOR GRAPH FOR MODULED OVER COMMUTATIVE RINGS112204810.22044/jas.2020.9194.1448ENKatayoun NozariDepartment of Mathematics, Imam Khomeini International University, P.O.Box 34149-16818, Qazvin, Iran.Sh. PayroviDepartment of Mathematics, Imam Khomeini International University, P.O.Box
34149-16818, Qazvin, Iran.Journal Article20191215Let $R$ be a commutative ring and $M$ be an $R$-module. The<br /> annihilator graph of $M$, denoted by $AG(M)$ is a simple undirected<br /> graph associated to $M$ whose the set of vertices is<br /> $Z_R(M) \setminus {\rm Ann}_R(M)$ and two distinct vertices $x$ and<br /> $y$ are adjacent if and only if ${\rm Ann}_M(xy)\neq {\rm<br /> Ann}_M(x) \cup {\rm Ann}_M(y)$. In this paper, we study the<br /> diameter and the girth of $AG(M)$ and we characterize all modules<br /> whose annihilator graph is complete. Furthermore, we look for the<br /> relationship between the annihilator graph of $M$ and its zero-divisor<br /> graph.https://jas.shahroodut.ac.ir/article_2048_120f446e61ae463a532c97a0913f6a36.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES1320204910.22044/jas.2020.9439.1460ENY. IraniDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
56199-11367, Ardabil, Iran.K. BahmanpourDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
56199-11367, Ardabil, Iran.Gh. GhasemiDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
56199-11367, Ardabil, Iran.Journal Article20200303Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $\mathrm{pd}_R\, A=\dim R$. Furthermore, as an application of these results it is shown that<br />$$\pdd H^i_{{\frak p}R_{\frak p}}(M_{\frak p}, N_{\frak p})\leq \pd H^{i+\dim R/{\frak p}}_{\frak m}(M,N)$$<br />for each ${\frak p}\in \mathrm{Spec} R$ and each integer $i\geq 0$. This result answers affirmatively a question raised by the present authors in [13].<br /><br />https://jas.shahroodut.ac.ir/article_2049_98e21e379801f667abcbe8f1b5044ff5.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901SOME PROPERTIES ON DERIVATIONS OF LATTICES2133205010.22044/jas.2020.7088.1347ENMayuka FKawaguchiGraduate School of Information Science and Technology, Hokkaido University,
P.O.Box 060-0814 Sapporo, JapanMichiro KONDODepartment of Mathematics, School of System Design and Technology, Tokyo Denki
University, P.O.Box 120-8551 Tokyo, JapanJournal Article20180517In this paper we consider some properties of derivations of lattices and show that (i) for a derivation $d$ of a lattice $L$ with the maximum element $1$, it is monotone if and only if $d(x) \le d(1)$ for all $x\in L$ (ii) a monotone derivation $d$ is characterized by $d(x) = x\wedge d(1)$ and (iii) simple characterization theorems of modular lattices and of distributive lattices are given by derivations. We also show that, for a distributive lattice $L$ and a monotone derivation $d$ of it, the set ${\rm Fix}_d(L)$ of all fixed points of $d$ is isomorphic to the lattice $L/\ker (d)$. We provide a counter example to the result (Theorem 4) proved in [3].https://jas.shahroodut.ac.ir/article_2050_521538d527cfd40d77ed799e48ec18d6.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901DEFICIENCY ZERO GROUPS IN WHICH PRIME POWER OF GENERATORS ARE CENTRAL3543205110.22044/jas.2020.9361.1456ENM. AhmadpourDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O.Box 56199-11367, Ardabil, Iran.H. AbdolzadehDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O.Box 56199-11367, Ardabil, Iran.Journal Article20200203The infinite family of groups defined by the presentation $G_p=\langle x, y|x^p=y^p,\; xyx^my^n=1\rangle$, in which $p$ is a prime in $\{2,3,5\}$ and $m,n\in\mathbb{N}_0$, will be considered and finite and infinite groups in the family will be determined. For the primes $p=2,3$ the group $G_p$ is finite and for $p=5$, the group is finite if and only if $m\equiv n\equiv1\pmod5$ is not the case.https://jas.shahroodut.ac.ir/article_2051_e39e91ea25ff14b11a5e14de63c7d0a3.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901C#-IDEALS OF LIE ALGEBRAS4551205310.22044/jas.2020.9458.1461ENL. GoudarziDepartment of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran.Journal Article20200311Let $L$ be a finite dimensional Lie algebra. A subalgebra $H$ of $L$ is called a $c^{\#}$-ideal of $L$, if there is an ideal $K$ of $L$ with $L=H+K$ and $H\cap K$ is a $CAP$-subalgebra of $L$. This is analogous to the concept of a $c^{\#}$-normal subgroup of a finite group. Now, we consider the influence of this concept on the structure of finite dimentional Lie algebras.<br /><br />https://jas.shahroodut.ac.ir/article_2053_33d2e7d3c4127015fe8fd3c756ef1561.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901GRAPHS WITH TOTAL FORCING NUMBER TWO, REVISITED5360205410.22044/jas.2020.9229.1451ENM. AlishahiFaculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box:
316-3619995161, Shahrood, Iran.0000-0001-6588-8520E. Rezaei-SaniFaculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box:
316-3619995161, Shahrood, Iran.Journal Article20191228A subset of the vertex set of a graph $G$ is called a zero forcing set if by considering them colored and, as far as possible, a colored vertex with exactly one non-colored neighbor forces its non-colored neighbor to get colored, then the whole vertices of $G$ become colored. The total forcing number of a graph $G$, denoted by $F_t(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a subgraph with no isolated vertex. The connected forcing number, denoted by $F_c(G)$, is the cardinality of a smallest zero forcing set of $G$ which induces a connected subgraph. In this paper, we first characterize the graphs with $F_t(G)=2$ and, as a corollary, we characterize the graphs with $F_c(G)=2$.https://jas.shahroodut.ac.ir/article_2054_f054a10538ec58a6a4782c9d7458d295.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901FUZZY MEDIAL FILTERS OF PSEUDO BE-ALGEBRAS6182205510.22044/jas.2020.9477.1464ENA. RezaeiDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697,
Tehran, Iran.0000-0002-6003-3993Journal Article20200317In this paper, the notion of fuzzy medial filters of a pseudo BE-algebra<br /> is defined, and some of the properties are investigated. We show that the<br /> set of all fuzzy medial filters of a pseudo BE-algebra is a complete lattice.<br /> Moreover, we state that in commutative pseudo BE-algebras fuzzy filters and<br /> fuzzy medial filters coincide. Finally, the notion of a fuzzy implicative filter is introduced<br /> and proved that every fuzzy implicative filter is a fuzzy medial filter, and we<br /> show that the converse is not valid in general.https://jas.shahroodut.ac.ir/article_2055_7e0029984b4c30549134146033ebb382.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901m-TOPOLOGY ON THE RING OF REAL-MEASURABLE FUNCTIONS83106205610.22044/jas.2020.9557.1470ENH. YousefpourFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.A. A. EstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.0000-0001-8993-5109A. Mahmoudi DarghadamFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.Gh. SadeghiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar,
Iran.Journal Article20200410In this article we consider the $m$-topology on <br /> \linebreak <br /> $M(X,\mathscr{A})$, the ring of all real measurable functions on a measurable space<br /> $(X, \mathscr{A})$, and we denote it by $M_m(X,\mathscr{A})$.<br /> We show that $M_m(X,\mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, \mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,\mathscr{A})\cong \mathbb R^n$ as topological rings. <br /> Also, we show that $M_m(X,\mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,\mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,\mathscr{A})= {M}_{u}(X,\mathscr{A})$, if and only if $ M_m(X,\mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,\mathscr{A})$ is a connected space, if and only if $M_m(X,\mathscr{A})$ is a locally connected space, if and only if $M^*(X,\mathscr{A})$ is a connected subset of $M_m(X,\mathscr{A})$.https://jas.shahroodut.ac.ir/article_2056_ea0b901cccd560776fdc7441db04840b.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901SOME RESULTS ON ϕ -(k,n)-CLOSED SUBMODULES107118205710.22044/jas.2020.8817.1426ENM. H. Moslemi KoupaeiDepartment of Mathematics, Roudehen Branch, Islamic Azad University , Roudehen,
Iran.Journal Article20190817Let $R$ be a commutative ring with identity and $M$ be a unitary $R$ -module. Let $S(M)$ be the set of all submodules of $M$ and $\phi :S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function. A proper submodule $N$ of $M$ is called $\phi$ -semi-$n$-absorbing if $r^{n} m\in N\setminus \phi(N)$ where $r\in R, m\in M$ and $n\in {\Bbb Z}^+$, then $r^{n} \in (N:M)$ or $r^{n-1} m\in N$. Let $k$ and $n$ are positive integers where $k>n$. <br />A proper submodule $N$ of $M$ is called $\phi$ -$(k,n)$- closed submodule, if $ r^{k}m\in N\setminus \phi(N)$ where $r\in R$, $m\in M$ and $k\in {\Bbb Z}^+$, then $r^{n}\in (N:M)$ or $r^{n-1}m\in N$. In this work, firstly, we will study some general results when we use the definition $\phi$ -$(k,n)$- closed submodule. Moreover, we prove main results of the $\phi$ -$(k,n)$- closed submodule for various modules.<br /><br />https://jas.shahroodut.ac.ir/article_2057_aa67030b0e9edac480d53bb3802e5ce1.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901ALGORITHMIC ASPECTS OF ROMAN GRAPHS119135205810.22044/jas.2020.8188.1400ENA. PoureidiFaculty of Mathematical Sciences, Shahrood University of Technology, Shahrood,
Iran.Journal Article20190311Let $G=(V, E)$ be a graph. <br /> A set $S \subseteq V$ is called a dominating set of $G$ if for every $v\in V-S$ there is at least one vertex $u \in N(v)$ such that $u\in S$. The domination number of $G$, denoted by $\gamma(G)$, is equal to the minimum cardinality of a dominating set in $G$.<br /> A Roman dominating function (RDF) on $G$ is a function $f:V\longrightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u$ with $f(u)=2$. The weight of $f$ is the sum $f(V)=\sum_{v\in V}f (v)$. The minimum weight of a RDF on $G$ is the Roman domination number of $G$, denoted by $\gamma_R(G)$.<br /> A graph $G$ is a Roman Graph if<br /> $\gamma_R(G)=2\gamma(G)$.<br /> <br /> <br /> In this paper, we first study the complexity issue of the problem posed<br /> in [E.J. Cockayane, P.M. Dreyer Jr., S.M. Hedetniemi and S.T. Hedetniemi, On Roman domination in graphs, \textit{Discrete Math.} 278 (2004), 11--22], and show that the problem of deciding whether a given graph is a Roman graph is NP-hard even when restricted<br /> to chordal graphs. Then, we give linear algorithms that compute the domination number and <br /> the Roman domination number of a given unicyclic graph. Finally, using these algorithms we give a linear algorithm that decides whether a given unicyclic graph is a Roman graph.https://jas.shahroodut.ac.ir/article_2058_3654a9bd6949d46a357a1413df58873e.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901LEFT ABSORBING HYPER K-ALGEBRAS137149205910.22044/jas.2020.8717.1423ENS. MadadiDepartment of Mathematics, University of SHAHED, Tehran, Iran.M. A. Nasr-AzadaniDepartment of Mathematics, University of SHAHED, P.O.Box 18151-159, Tehran,
Iran.Journal Article20190721In the present manuscript, we introduce a type of hyper K-algebra which is called left absorbing hyper K-algebra and investigate some of the related properties. We also show that set of all types of positive implicative and commutative hyper K-ideal form a distributive latttice and study their diagrams when positive implicative and commutative hyper K-ideal are a hyper K-ideal and the hyper K-algebra is left absorbing.https://jas.shahroodut.ac.ir/article_2059_f6f83e8f6ad98309bef297966deface2.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901ON GP-FLATNESS PROPERTY151174206110.22044/jas.2020.8923.1437ENH. Mohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran.M. AbbasiDepartment of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan,
Iran.Journal Article20190926It is well-known that, using principal weak flatness property, some important monoids are characterized, such as regular monoids, left almost regular monoids, and so on. In this article, we recall a generalization of principal weak flatness called GP-flatness, and characterize monoids by this property of their right (Rees factor) acts. Also we investigate GP coherent monoids.https://jas.shahroodut.ac.ir/article_2061_8c60a6b1d34ef82207bc7be5f7bfb0c7.pdf