University of GuilanJournal of Algebra and Related Topics2345-393110120220601S-small and S-essential submodules110554910.22124/jart.2021.20856.1328ENS.RajaeeDepartment of Mathematics, University
of Payame Noor,Tehran, Iran.0000-0001-8244-0752Journal Article20211017This paper is concerned with S-comultiplication modules which are a generalization of comultiplication modules.<br />In section 2, we introduce the S-small and S-essential submodules of a unitary $R$-module $M$ over a commutative ring $R$ with $1\neq 0$ such that S is a multiplicatively closed subset of $R$. We prove that if $M$ is a faithful S-strong comultiplication $R$-module and $N\ll ^{S}M$, then there exist an ideal $I\leq ^{S}_{e}R$ and an $t\in S$ such that $t(0 :_{M}I)\leq N\leq (0 :_{M}I)$. The converse is true if $S\subseteq {\rm U}(R)$ such that ${\rm U}(R)$ is the set of all units of $R$. Also, we prove that if $M$ is a torsion-free S-strong comultiplication module, then $N\leq ^{S}_{e} M$ if and only if there exist an ideal $I\ll ^{S}R$ and an $s\in S$ such that $s(0 :_{M} I)\leq N\leq (0 :_{M} I)$. In section 3, we introduce the concept of S-quasi-copure submodule $N$ of an $R$-module $M$ and investigate some results related to this class of submodules.https://jart.guilan.ac.ir/article_5549_7957f2a8cc070d865a0b05be50a2dfbe.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601Generalization of $n$-ideals1133561410.22124/jart.2022.19789.1282ENS.KarimzadehDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, , Rafsanjan, Iran.0000-0002-8395-2626S.HadjirezaeiDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, , Rafsanjan, Iran.0000-0002-8994-5523Journal Article20210606Let f:A→B be a ring homomorphism and let J<br />be an ideal of B. We proved some results concerning n-ideals and<br />(2; n)-ideals of A⋈^f J. Then we recall a proper ideal I of A as √(δ(0))-ideal if ab ϵ I then b ∈ I or a ∈ √(δ(0)) for every a; b ∈ A. We investigate some properties of √(δ(0))-ideal with similar n-ideals<br />and J-ideals.https://jart.guilan.ac.ir/article_5614_0b2f1094c89e2b9611a034b358486a88.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601Mappings between the lattices of varieties of submodules3550561510.22124/jart.2021.19574.1272ENH.Fazaeli MoghimiDepartment of Mathematics, University of Birjand, Birjand, Iran.M.NoferestiDepartment of Mathematics, University of Birjand, Birjand, Iran.Journal Article20210509Let $R$ be a commutative ring with identity and $M$ be an $R$-module. It is shown that the usual lattice $\mathcal{V}(_{R}M)$ of varieties of submodules of $M$ is a distributive lattice. If $M$ is a semisimple $R$-module and the unary operation $^{\prime}$ on $\mathcal{V}(_{R}M)$ is defined by $(V(N))^{\prime}=V(\tilde{N})$, where $M=N\oplus \tilde{N}$, then the lattice $\mathcal{V}(_{R}M)$ with $^{\prime}$ forms a Boolean algebra. In this paper, we examine the properties of certain mappings between $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$, in particular considering when these mappings are lattice homomorphisms. It is shown that if $M$ is a faithful primeful $R$-module, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic lattices, and therefore $\mathcal{V}(_{R}M)$ and the lattice $\mathcal{R}(R)$ of radical ideals of $R$ are anti-isomorphic lattices. Moreover, if $R$ is a semisimple ring, then $\mathcal{V}(_{R}R)$ and $\mathcal{V}(_{R}M)$ are isomorphic Boolean algebras, and therefore $\mathcal{V}(_{R}M)$ and $\mathcal{L}(R)$ are anti-isomorphic Boolean algebras.https://jart.guilan.ac.ir/article_5615_840b9b5fe3c86c4b2227a99c740e6ab7.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601Generalized orthogonal graphs of characteristic a power of 25161561610.22124/jart.2021.20476.1310ENS.SriwongsaDepartment of Mathematics, King Mongkut's University of Technology Thonburi, Bangkok, Thailand.Y.WeiSchool of Mathematics and Statistics, Nanning Normal University, Nanning , P.R. ChinaJournal Article20210829Let $R$ be a finite local ring of characteristic a power of $2$ with the residue field $k$.<br />In this paper, we define a generalized orthogonal graph on a module of rank at least $2$ over $R$. Then we study its graph properties via the same graph over $k$. The number of vertices and the valency of each vertex in this graph over $R$ are computed. We also prove that this graph is arc transitive and find its diameter. Moreover, the first subconstituent of this orthogonal graph is considered. We show that it is a generalized strongly regular graph.https://jart.guilan.ac.ir/article_5616_290a8d2bfedf137302f52ac9945498b5.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601*-alpha-derivation on prime *-rings6369561710.22124/jart.2021.20488.1312ENK.KumarDepartment of Mathematics,
Aligarh Muslim University, Aligarh, India11orcid.org1000000003-437-8775Journal Article20210830Let $\Re$ be an associative ring with involution $*$. An additive map $\lambda\rightarrow \lambda^{*}$ of $\Re$ into itself is called an involution if the following conditions are satisfied $(i) (\lambda\mu)^{*}=\mu^{*}\lambda^{*}$, $(ii) (\lambda^{*})^{*}=\lambda ~~ \mbox{for all}~ \lambda,\mu\in \Re$. A ring equipped with an involution is called an $*$-ring or ring with involution. The aim of the present paper is to establish some results on $*$-$\alpha$-derivations in $*$-rings and investigate the commutativity of prime $*$-rings admitting $*$-$\alpha$-derivations on $\Re$ satisfying certain identities also prove that if $\Re$ admits a reverse $*$-$\alpha$-derivation $\delta$ of $\Re$, then $\alpha\in Z(\Re)$ and some related results have also been discussed.https://jart.guilan.ac.ir/article_5617_0b1e334b2f0a69d28e7cc1c2379b8326.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601A new radical in free modules7194561810.22124/jart.2022.20214.1295ENP.AdhamiDepartment of Mathematics, Hormozgan University, Bandar Abbas, Iran.J.MoghaderiDepartment of Mathematics, Hormozgan University, Bandar Abbas, Iran.0000-0003-2217-5461Journal Article20210722An $R$-module $M$ is called torsion-free, if $rx=0$ for $r\in R$ and $x\in M$ implies that $r=0$ or $x=0$. In this paper, we introduce the notions semi torsion-free modules and quasi torsion-free modules. We show that a submodule $N$ of an $R$-module $M$ is a $P$-primary submodule if and only if $\dfrac{R}{P}$-module $\dfrac{M}{N}$ is semi torsion-free. Also we define a new radical in free modules and find some characterizations of it. We prove that for $P$-submodule $N$ of a free $R$-module $F$ which $\sqrt N \subsetneqq F$, we have for any $r \in R$ and $m \in F$, $rm \in N$ implies $r \in \sqrt P$ or $m \in \sqrt N$ if and only if $\dfrac{R}{P}$-module $\dfrac{F}{N}$ is quasi torsion free.https://jart.guilan.ac.ir/article_5618_d7c8d88953c204a9d9e51f962d4ba772.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601On Dominions and Determination of Closed Varieties of Semigroups95111530210.22124/jart.2021.19128.1262ENS.AbbasDepartment of mathematics, Aligarh Muslim University, Aligarh-202002, India.W.AshrafDepartment of mathematics, Aligarh Muslim University, Aligarh-202002, India.Journal Article20210310 It is known that all subvarieties of variety of all semigroups are not absolutely closed. So, it is a natural question to find out those subvarieties of variety of all semigroups that are closed in itself or close in larger subvarieties of variety of all semigroups. We have gone through this open problem and able to determine some closed varieties of semigroups defined by the identities $axy=yxax~[axy=xyxa]$ and $axy=yxxa$ by using Isbell's zigzag theorem as an essential tool. Further, we partially generalize a result of Isbell on semigroup dominions from the class of commutative semigroups to some generalized classes of commutative semigroups by showing that dominions of such semigroups belong to the same class.https://jart.guilan.ac.ir/article_5302_5db8e943993d1f13e42c75e72c463a2f.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601Jordan algebra bundles and Jordan Rings113118561910.22124/jart.2021.19722.1278ENR.KumarDepartment of Mathematics, School of applied sciences, Reve University, Bengaluru, India.Journal Article20210526In this paper, We define Jordan algebra bundles of finite type and we give one-one correspondence between Jordan algebra bundles of finite type and Jordan rings.https://jart.guilan.ac.ir/article_5619_d6f2d40bf33586fbce5ed824c1df6651.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601Some results on domination in the generalized total graph of a commutative ring119128562010.22124/jart.2021.19238.1265END.PatwariDepartment of Mathematics, Gauhati University, Guwahati
781014, Assam, IndiaH. K.SaikiaDepartment of Mathematics, Gauhati University, Guwahati-781014, Assam, IndiaJ.GoswamiDepartment of Mathematics, Gauhati University, Guwahati-781014, Assam, India0000-0002-1786-752XJournal Article20210326Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R/H is a saturated multiplicatively closed subset of R. Anderson and Badawi [4] introduced the generalized total graph of R as an undirected simple graph GTH(R) with vertex set as R and any two distinct vertices x and y are adjacent if and only if x + y ϵ H. The main objective of this paper is to study the domination properties of the graph GTH(R). We determine the domination number of GTH(R) and its induced subgraphs GTH(H) and GTH(R/H). We establish a relationship between<br />the domination number of GTH(R) and the same of GTH(R/H). We also establish a relationship between diameter and domination number of GTH(R/H). In addition,we obtain the bondage number of GTH(R). Finally, a relationship between girth and bondage number of GTH(R/H) has been established.https://jart.guilan.ac.ir/article_5620_4c93f7555f7e38654ce4629b47a2f3da.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220620Integral closure of a filtration relative to a Noetherian module129141562110.22124/jart.2021.20320.1302ENF.DorostkarDepartment of Pure Mathematics, University of Guilan, Rasht, IranM.Yahyapour-DakhelDepartment of Pure Mathematics, University of Guilan, Rasht, IranJournal Article20210807 Let $M$ be a Noetherian $R-$module.<br />In this paper we will introduce the integral closure of a filtration ${\mathcal{F}}=\{I_{n}\}_{n\geq 0}$ relative to the Noetherian module $M$ and prove some related results.\\<br /> The integral closure of a filtration ${\mathcal{F}}=\{I_{n}\}_{n\geq 0}$ relative to $M$ is a filtration and it has an interesting relationship with the integral closure of the filtration<br /> ${\widetilde{\mathcal{F}}}=\{\widetilde{I}_{n}\}_{n\geq 0}$, where $\widetilde{I}_{n}$ is the image of $I_n$ under the natural ring homomorphism $R\rightarrow R/(Ann_R(M))$ for every $n\geq 0$.https://jart.guilan.ac.ir/article_5621_670999f57a2e5182656b4bc47f990dba.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601On Special Amalgams and Closed Varieties of Posemigroups143158562210.22124/jart.2021.20345.1303ENS. A.AhangerDepartment of Mathematics, Central University of Kashmir, , Ganderbal, IndiaS.BanoDepartment of Mathematics Central University of Kashmir, Ganderbal, India0000-0002-3577-7771A. H.ShahDepartment of Mathematics, Central University of Kashmir, , Ganderbal, IndiaJournal Article20210810In this paper we extend a result of Scheiblich by showing that variety of po-normal bands is closed. We also extend the well known results to posemigroups namely, that pogroups and inverse posemigroups have special amalgamation property in the category of all posemigroups and commutative posemigroups, respectively. Finally, we find some varieties of posemigroups which are closed if they are self convex.https://jart.guilan.ac.ir/article_5622_71c3ec91e292acae17721021b90342fa.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-393110120220601A note on generalized derivations on prime ideals159169562310.22124/jart.2021.20131.1291ENN.RehmanDepartment of Mathematics
Aligarh Muslim University
Aligarh, India0000-0003-3955-7941H. M.AlnoghashiDepartment of Mathematics
Aligarh Muslim University
Aligarh, IndiaM.HonganDepartment of Mathematics, Seki 772, Maniwa, Okayama, Japan.Journal Article20210712This paper investigates the structure of the quotient ring $\mathscr{R}/ \mathscr{P}$, where $\mathscr{R}$ is an arbitrary ring and $\mathscr{P}$ is a prime ideal of $\mathscr{R}$ . We show that the structure of this class of rings has a relationship with the behaviour of generalized derivations satisfying algebraic identities involving prime ideals.https://jart.guilan.ac.ir/article_5623_4b2731c22ceb4a0ffddd7b40329ac424.pdf