Shahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101ON THE COMPUTATIONAL COMPLEXITY ASPECTS OF PERFECT ROMAN DOMINATION189202246910.22044/jas.2021.11146.1554ENS.H.MirhoseiniDepartment of Mathematics, Shahed University, Tehran, Iran.N.Jafari RadDepartment of Mathematics, Shahed University, Tehran, Iran.Journal Article20210827A perfect Roman dominating function (PRDF) on a graph $G$ is a function $ f:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF $f$ is the sum of the weights of the vertices under $f$. The perfect Roman domination number of $G$ is the minimum weight of a PRDF in $G$. In this paper we study algorithmic and computational complexity aspects of the minimum perfect Roman domination problem (MPRDP). We first correct the proof of a result published in [Bulletin<br />Iran. Math. Soc. 14(2020), 342--351], and using a similar argument, show that MPRDP is APX-hard for graphs with bounded degree 4.<br />We prove that the decision problem associated to MPRDP is NP-complete even when restricted to star convex bipartite graphs. Moreover, we show that MPRDP is solvable in linear time for bounded tree-width<br />graphs. We also show that the perfect domination problem and perfect Roman domination problem are not equivalent in computational complexity aspects. Finally we propose an integer linear programming formulation for MPRDP.https://jas.shahroodut.ac.ir/article_2469_24a6eab2d33ba5f82efa838562f8f257.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101r-CLEAN RINGS RELATIVE TO RIGHT IDEALS203224247010.22044/jas.2021.10627.1525ENH. IbrahimHakmiDepartment of Mathematics, Damascus University, Damascus, Syria.orcid.org/0000-0002-4583-2009B. AliAlusseinDepartment of Mathematics, Damascus University, Damascus, Syria.Journal Article20210310Abstract.An associative ring R with identity is called r¡clean ring if every<br />element of R is the sum of a regular and an idempotent element. In this paper,<br />we introduce the concept of r-clean rings relative to right ideal. We study<br />various properties of these rings. We give some relations between r-clean<br />rings and r-clean rings of 2 2 matrices over R relative to some right ideal<br />P. New characterization obtained include necessary and sufficient conditions<br />of a ring R to be r-clean in terms of P-regular, P-local and P-clean rings.<br />Also, We prove that every ring is r-clean relative to any maximal right ideal<br />of it.https://jas.shahroodut.ac.ir/article_2470_45b586cbdfdebd0f60d642784ff46ddd.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101GRADED I-PRIME SUBMODULES225243247110.22044/jas.2022.11158.1556ENI.AkrayDepartment of Mathematics, Soran University, Erbil, Iraq.Sh.OthmanDepartment of Mathematics, Salahaddin university, Erbil, Iraq.A.JabbarDepartment of Mathematics, University of Sulaimani, Erbil, Iraq.H.HusseinDepartment of Mathematics, Soran University, Erbil, Iraq.Journal Article20210901Let $R= \bigoplus_{g \in G} R_g$ be a $G-$graded commutative ring with identity, $I$ be a graded ideal and let $M$ a $G-$graded unitary $R$-module, where $G$ is a semigroup with identity $e$. We introduce graded $I-$prime ideals (submodules) as a generalizations of the classical notions of prime ideals (submodules). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module $F$, a graded submodule $P$ of $M$ is $I-$prime if and only if $F \otimes P$ is graded $I-$prime submodule of $F \otimes M$. As an application, for finitely generated graded module $M$ over Noetherian graded ring $R$, the completion of graded $I-$prime submodules is $I-$prime submodule.https://jas.shahroodut.ac.ir/article_2471_13439e0076a30d32464ae850e203bbbb.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101FALTINGS’ LOCAL-GLOBAL PRINCIPLE FOR THE MINIMAXNESS OF LOCAL COHOMOLOGY MODULES DEFINED BY A SYSTEM OF IDEALS245258247210.22044/jas.2022.10587.1524ENF.Dehghani-ZadehDepartment of Mathematics, Islamic Azad University, Yazd branch, Yazd, Iran.0000-0003-1244-1164A.R.HajikarimiDepartment of Mathematics, Mobarakeh Branch,
Islamic Azad University, Isfahan, Iran.Journal Article20210227Let R be a commutative Noetherian ring with nonzero identity. Let φ be a system of ideals of R and let M, N two finitely generated R-modules. We prove that there are local- global principles for the finiteness and minimaxness of generalized local cohomology module H_φ^i (M, N) , in certain cases.https://jas.shahroodut.ac.ir/article_2472_fef60e2ba9df3b64a6f730f9cfb88519.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101ON HOMOLOGICAL CLASSIFICATION OF MONOIDS BY CONDITION (PWPsc) OF RIGHT ACTS259283247310.22044/jas.2022.11070.1548ENHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranLeilaNouriDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran0000-0002-2240-583xJournal Article20210807In this paper, we introduce Condition (PWPsc) as a generalization of Condition (PWP_E) of acts over monoids, and we observe that Condition (PWPsc) does not imply Condition (PWP_E). In general, we show that Condition (PWPsc) implies the property of being principally weakly flat, and that in left PSF<br />monoids, the converse of this implication is also true. Moreover, we present some general properties and a homological classification of monoids by comparing Condition (PWPsc) with some other properties. Finally, we describe left PSF monoids for which S^I_S satisfies Condition (PWPsc) for any nonempty set I.https://jas.shahroodut.ac.ir/article_2473_1b05bdde687a665b7252ff249dc0f62c.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101INTUITIONISTIC FALLING SHADOWS APPLIED TO COMMUTATIVE IDEALS IN BCK-ALGEBRAS285297247410.22044/jas.2022.10104.1503ENR. A.BorzooeiDepartment of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti
University, Tehran, Iran.0000-0001-7538-7885X. L.XinSchool of Mathematics, Northwest University, P.O. Box 710127, Xi’an, China.Y. B.JunDepartment of Mathematics Education, Gyeongsang National University, P.O. Box
52828, Jinju, Korea.0000-0002-0181-8969Journal Article20200925The notion of commutative falling intuitionistic fuzzy ideal of a BCK-algebra is introduced and related properties are investigated. We verify that every commutative intuitionistic fuzzy ideal is a commutative falling intuitionistic fuzzy ideal, and provide example to show that a commutative falling intuitionistic fuzzy ideal is not a commutative intuitionistic fuzzy ideal. Relations between a falling intuitionistic fuzzy ideal and a commutative falling intuitionistic fuzzy ideal are considered, and a condition for a falling intuitionistic fuzzy ideal to be a commutative falling intuitionistic fuzzy ideal is provided.https://jas.shahroodut.ac.ir/article_2474_6acb8cf9a78a23653972117b998a766e.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101ON DETERMINING THE DISTANCE SPECTRUM OF A CLASS OF DISTANCE INTEGRAL GRAPHS299308247510.22044/jas.2022.11207.1559ENSeyed M.MirafzalDepartment of Mathematics, Lorestan University, Khorramabad, Iran.R.KoganiDepartment of Mathematics, Lorestan University, Khorramabad, Iran.Journal Article20210914The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix<br />$D(G)$. A graph is called distance integral if all of its<br />distance eigenvalues are integers.<br />Let $n$ and $k$ be integers with $n>2k, k\geq1$. The bipartite Kneser graph $H(n,k)$ is the graph with the set of all $k$ and $n-k$ subsets of the set $[n]=\{1,2,...,n\}$ as vertices, in which two vertices are adjacent if and only if one of them is a subset of the other. <br />In this paper, we determine the distance spectrum of $H(n,1)$. Although the obtained result is not new \cite{12}, but our proof is new. The main tool that we use in our work is the orbit partition method in algebraic graph theory for finding the eigenvalues of graphs. We introduce a new method for<br />determining the distance spectrum of $H(n,1)$ and show how<br />a quotient matrix can contain all distance eigenvalues of<br />a graph.https://jas.shahroodut.ac.ir/article_2475_937ff50362e356b908acfb15d144399f.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101ON THE PATH HYPEROPERATION AND ITS CONNECTIONS WITH HYPERGRAPH THEORY309321247610.22044/jas.2022.11493.1580ENR.Bayat TajvarMathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU)
University, Tehran, Iran.M.LatifiMathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU)
University, Tehran, Iran.Journal Article20211215In this paper, we introduce a path hyperoperation associated with a hypergraph,<br />which is an extension of the Corsini’s hyperoperation.<br />We investigate some related properties and study relations between<br />the path hyperoperation and hypergraph theory.https://jas.shahroodut.ac.ir/article_2476_babf1fffa89f5466930ed0ca5e5712c7.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101A NOTE ON Cc(X) VIA A TOPOLOGICAL RING323334247710.22044/jas.2022.11467.1579ENR.MohamadianDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.000000033350366XM.NamdariDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.0000-0003-0966-7234H.NajafianDepartment of Mathematics, Shahid Chamran University of Ahvaz, P.O. Box
6135783151, Ahvaz, Iran.S.SoltanpourDepartment of Science, Petroleum University of Technology, P.O. Box 6318714317,
Ahvaz, Iran.0000-0002-1072-9845Journal Article20211209Let $C_c(X)$ (resp., $C_c^*(X)$) denote the functionally<br />countable subalgebra of $C(X)$ (resp., $C^*(X)$),<br />consisting of all functions (resp., bounded functions) with countable image.<br />$C_c(X)$ (resp., $C_c^*(X)$) as a topological ring via $m_c$-topology (resp., $m^*_c$-topology) and $u_c$-topology (resp., $u^*_c$-topology) is investigated and the equality of the latter two topologies is characterized. <br />Topological spaces which are called $N$-spaces are introduced and studied.<br />It is shown that the $m_c$-topology on $C_c(X)$ and its relative topology as a subspace of $C(X)$ (with $m$-topology) coincide if and only if $X$ is an $N$-space. We also show that $X$ is pseudocompact if and only if it is both a countably pseudocompact, and an $N$-space.https://jas.shahroodut.ac.ir/article_2477_e924e7f0f47be03484e4067a481fe8a8.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101PERFECTNESS OF THE ANNIHILATOR GRAPH OF ARTINIAN COMMUTATIVE RINGS335343247810.22044/jas.2022.11358.1571ENM.AdlifardDepartment of Mathematics, Roudbar Branch, Islamic Azad University, Roudbar,
Iran.Sh.PayroviDepartment of Mathematics, Imam Khomeini International University, P.O. Box
34149-1-6818, Qazvin, Iran.Journal Article20211103Let $R$ be a commutative ring and $Z(R)$ be the set of its zero-divisors.<br />The annihilator graph of $R$, denoted by $AG(R)$ is a simple undirected graph whose vertex<br />set is $Z(R)^*$, the set of all nonzero zero-divisors of $R$, and two distinct vertices $x$ and<br />$y$ are adjacent if and only if ${\rm ann}_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$.<br />In this paper, perfectness of the annihilator graph for some classes of rings is investigated.<br />More precisely, we show that if $R$ is an Artinian ring, then $AG(R)$ is perfect.https://jas.shahroodut.ac.ir/article_2478_e39eb81cb204b26ca17a13df1e6c0f32.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101A GRAPH ASSOCIATED TO FILTERS OF A LATTICE345359247910.22044/jas.2022.10633.1526ENSh.Ebrahimi AtaniDepartment of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran.M.KhoramdelDepartment of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran.S.Dolati Pish HesariDepartment of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran.M.Nikmard RostamalipourDepartment of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran.Journal Article20210311Let $L$ be a lattice with the least element $0$ and the greatest element $1$. In this paper, we associate a graph to filters of $L$, in which the vertex set is being the set of all non-trivial filters of $L$, and two distinct vertices $F$ and $E$ are adjacent if and only if $F \cap E \neq \{1\}$. We denote this graph by $\mathcal{G}$ $(L)$. The basic properties and possible structures of $\mathcal{G}$ $(L)$ are studied. Moreover, we characterize the planarity of $\mathcal{G}$ $(L)$.https://jas.shahroodut.ac.ir/article_2479_32018ff0f5780e323702b204f0d79d44.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-512810220230101WEAKLY BAER RINGS361374248010.22044/jas.2022.11148.1555ENS.MehralinejadianDepartment of Mathematics, Central Tehran Branch, Islamic Azad University,
Tehran, Iran.0000-0003-3145-5772A.MoussaviDepartment of Mathematics, Central Tehran Branch, Islamic Azad University,
Tehran, Iran.Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares
University, P.O. Box 14115-134, Tehran, Iran.Sh.SahebiDepartment of Mathematics, Central Tehran Branch, Islamic Azad University,
Tehran, Iran.Journal Article20210828We say a ring R with unity is left weakly Baer if the left annihilator<br />of any nonempty subset of R is right s-unital by right semicentral idempotents,<br />which implies that R modulo the left annihilator of any nonempty subset is<br />ﬂat. It is shown that, unlike the Baer or right PP conditions, the weakly<br />Baer property is inherited by polynomial extensions. Examples are provided<br />to explain the results.https://jas.shahroodut.ac.ir/article_2480_5a6c89e16b2af5f2297eb048bfa8d252.pdf