University of GuilanJournal of Algebra and Related Topics2345-39316220181201Classical Zariski Topology on Prime Spectrum of Lattice Modules114332610.22124/jart.2018.11106.1112ENV.BorkarDepartment of Mathematics, Yeshwant Mahavidyalaya, Nanded, IndiaP.GiraseDepartment of Mathematics, K K M College, Manwath, Dist- Parbhani. 431505. Maharashtra, India.0000-0002-7847-8296N.PhadatareDepartment of Mathematics, Savitribai Phule Pune University, Pune. Maharashtra. IndiaJournal Article20180817Let $M$ be a lattice module over a $C$-lattice $L$. Let $Spec^{p}(M)$ be the collection of all prime elements of $M$. In this article, we consider a topology on $Spec^{p}(M)$, called the classical Zariski topology and investigate the topological properties of $Spec^{p}(M)$ and the algebraic properties of $M$. We investigate this topological space from the point of view of spectral spaces. By Hochster's characterization of a spectral space, we show that for each lattice module $M$ with finite spectrum, $Spec^{p}(M)$ is a spectral space. Also we introduce finer patch topology on $Spec^{p}(M)$ and we show that $Spec^{p}(M)$ with finer patch topology is a compact space and every irreducible closed subset of $Spec^{p}(M)$ (with classical Zariski topology) has a generic point and $Spec^{p}(M)$ is a spectral space, for a lattice module $M$ which has ascending chain condition on prime radical elements.https://jart.guilan.ac.ir/article_3326_5f999e1eaeb83e79b53a441a2df5103f.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316220181201On the ranks of certain semigroups of order-preserving partial isometries of a finite chain1533332710.22124/jart.2019.10875.1109ENB.AliDepartment of Mathematics, Nigeria Defence Academy, Kano, NigeriaM. A.JadaDepartment of Mathematics, Bayero University Kano, Kano, NigeriaM. M.ZubairuDepartment of Mathematics, Faculty of Physical Sciences, Bayero University, Kano, Nigeria0000-0001-5099-5956Journal Article20180719Let $X_n={1,2,ldots,n}$ be a finite chain, $mathcal{ODP}_{n}$ be the semigroup of order-preserving partial isometries on $X_n$ and $N$ be the set of all nilpotents in $mathcal{ODP}_{n}$. In this work, we study the nilpotents in $mathcal{ODP}_{n}$ and investigate the ranks of two subsemigroups of $mathcal{ODP}_{n}$; the nilpotent generated<br />subsemigroup $langle Nrangle$ and the subsemigroup ~$L(n,r)= { alpha in mathcal{ODP}_{n} : |im~alpha|leq r}$.https://jart.guilan.ac.ir/article_3327_56a79b622398bebb17cc02cf3f649d3b.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316220181201Some results on a subgraph of the intersection graph of ideals of a commutative ring3561332810.22124/jart.2018.11188.1114ENS.VisweswaranDepartment of Mathematics,
Saurashtra University, Rajkot, India.P.VadhelDepartment of Mathematics,
Saurashtra University, Rajkot, IndiaJournal Article20180831The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. Let us denote the collection of all proper ideals of $R$ by $mathbb{I}(R)$ and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$. With $R$, we associate an undirected graph denoted by $g(R)$, whose vertex set is $mathbb{I}(R)^{*}$ and distinct vertices $I_{1}, I_{2}$ are adjacent in $g(R)$ if and only if $I_{1}cap I_{2}neq I_{1}I_{2}$. The aim of this article is to study the interplay between the graph-theoretic properties of $g(R)$ and the ring-theoretic properties of $R$.https://jart.guilan.ac.ir/article_3328_37da989245b3ff3ca164523e990de30b.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316220181201Basis of a multicyclic code as an Ideal in F[X_1,...,X_s]/<X_1^rho_1,...,X_s^rho_s>6378332910.22124/jart.2018.10977.1110ENR.AndriamifidisoaDepartment of Mathematics and Computer Science, University of Antananarivo, Antananarivo, MadagascarR. M.LalasoaDepartment of
Mathematics, University
of Antananarivo, Antananarivo, MadagascarT. J.RabeherimananaDepartment of
Mathematics, University
of Antananarivo, Antananarivo, MadagascarJournal Article20180801First, we apply the method presented by Zahra Sepasdar in the two-dimensional case to construct a basis of a three dimensional cyclic code. We then generalize this construction to a general $s$-dimensional cyclic code.https://jart.guilan.ac.ir/article_3329_8d8f5e7ba61fd23caa4b4d6f970ceb64.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316220181201Quasi-bigraduations of Modules, criteria of generalized analytic independence7996333010.22124/jart.2018.11137.1113ENY. M.DiaganaLaboratoire Math$acute{e}$matiques-Informatique, Universit$acute{e}$ Nangui Abrogoua, Abidjan, C$hat{o}$te d'IvoireJournal Article20180824Let $mathcal{R}$ be a ring. For a quasi-bigraduation $f=I_{(p,q)}$<br />of ${mathcal{R}} $ we define an $f^{+}-$quasi-bigraduation of an ${%<br />mathcal{R}}$-module ${mathcal{M}}$ by a family $g=(G_{(m,n)})_{(m,n)in<br />left(mathbb{Z}times mathbb{Z}right) cup {infty }}$ of subgroups of $%<br />{mathcal{M}}$ such that $G_{infty }=(0) $ and $I_{(p,q)}G_{(r,s)}subseteq<br />G_{(p+r,q+s)},$ for all $(p,q)$ and all $(r,s)in left(mathbb{N} times<br />mathbb{N}right) cup {infty }.$<br /> Here we show that $r$ elements of ${mathcal{R}}$ are $J-$independent of<br />order $k$ with respect to the $f^{+}$quasi-bigraduation $g$ if and only if<br />the following two properties hold: they are $J-$independent of order $k$ with respect to the $^+$%<br />quasi-bigraduation of ring $f_2(I_{(0,0)},I)$ and there exists a relation of<br />compatibility between $g$ and $g_{I}$, where $I$ is the sub-$mathcal{A}-$%<br />module of $mathcal{R}$ constructed by these elements. We also show that criteria of $J-$independence of compatible<br />quasi-bigraduations of module are given in terms of isomorphisms of graded<br />algebras.https://jart.guilan.ac.ir/article_3330_ee9644cecd6586d3400f22645cd4623f.pdfUniversity of GuilanJournal of Algebra and Related Topics2345-39316220181201Prime extension dimension of a module97106333110.22124/jart.2018.11232.1116ENT.DuraivelDepartment of Mathematics, Pondicherry University, Puducherry, India.S.MangayarcarassyDepartment of Mathematics, Pondicherry Engineering College, Puducherry, India.K.PremkumarDepartment of Mathematics, Indira Gandhi Institute of Technology, Odisha, India.Journal Article20180905We have that for a finitely generated module $M$ over a Noetherian ring $A$ any two RPE filtrations of $M$ have same length.<br /> We call this length as prime extension dimension of $M$ and denote it as $mr{pe.d}_A(M)$.<br /> This dimension measures how far a module is from torsion freeness. We show for every submodule (N) of (M), (mr{pe.d}_A(N)leqmr{pe.d}_A(M)) and (mr{pe.d}_A(N)+mr{pe.d}_A(M/N)geqmr{pe.d}_A(M)). We compute the prime<br /> extension dimension of a module using the prime extension dimensions of its primary submodules which occurs in a minimal primary decomposition of (0) in (M).https://jart.guilan.ac.ir/article_3331_8a5342e66d28b4c73dccff01968afa06.pdf