University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
Classical Zariski Topology on Prime Spectrum of Lattice Modules
1
14
EN
V.
Borkar
Department of Mathematics, Yeshwant Mahavidyalaya, Nanded, India
borkarvc@gmail.com
P.
Girase
0000-0002-7847-8296
Department of Mathematics, K K M College, Manwath, Dist- Parbhani. 431505. Maharashtra, India.
pgpradipmaths22@gmail.com
N.
Phadatare
Department of Mathematics, Savitribai Phule Pune University, Pune. Maharashtra. India
a9999phadatare@gmail.com
10.22124/jart.2018.11106.1112
Let $M$ 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.
prime element,prime spectrum,classical Zariski topology,finer patch topology
https://jart.guilan.ac.ir/article_3326.html
https://jart.guilan.ac.ir/article_3326_5f999e1eaeb83e79b53a441a2df5103f.pdf
University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
On the ranks of certain semigroups of order-preserving partial isometries of a finite chain
15
33
EN
B.
Ali
Department of Mathematics, Nigeria Defence Academy, Kano, Nigeria
bali@nda.edu.ng
M. A.
Jada
Department of Mathematics, Bayero University Kano, Kano, Nigeria
mrjaad@gmail.com
M. M.
Zubairu
0000-0001-5099-5956
Department of Mathematics, Faculty of Physical Sciences, Bayero University, Kano, Nigeria
mmzubairu.mth@buk.edu.ng
10.22124/jart.2019.10875.1109
Let $X_n=\{1,2,\ldots,n\}$ 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 $\langle N\rangle$ and the subsemigroup ~$L(n,r)= \{ \alpha \in \mathcal{ODP}_{n} : |im~\alpha|\leq r\}$.
Order-Preserving partial transformations,isometries,height and right (left) waist,idempotents and nilpotents
https://jart.guilan.ac.ir/article_3327.html
https://jart.guilan.ac.ir/article_3327_56a79b622398bebb17cc02cf3f649d3b.pdf
University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
Some results on a subgraph of the intersection graph of ideals of a commutative ring
35
61
EN
S.
Visweswaran
Department of Mathematics,
Saurashtra University, Rajkot, India.
s_visweswaran2006@yahoo.co.in
P.
Vadhel
Department of Mathematics,
Saurashtra University, Rajkot, India
pravin_2727@yahoo.com
10.22124/jart.2018.11188.1114
The 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 $R$.
Artinian ring,Special principal ideal ring,diameter,girth,clique number
https://jart.guilan.ac.ir/article_3328.html
https://jart.guilan.ac.ir/article_3328_37da989245b3ff3ca164523e990de30b.pdf
University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
Basis of a multicyclic code as an Ideal in F[X_1,...,X_s]/
63
78
EN
R.
Andriamifidisoa
Department of Mathematics and Computer Science, University of Antananarivo, Antananarivo, Madagascar
rmw278@yahoo.fr
R. M.
Lalasoa
Department of
Mathematics, University
of Antananarivo, Antananarivo, Madagascar
larissamarius.lm@gmail.com
T. J.
Rabeherimanana
Department of
Mathematics, University
of Antananarivo, Antananarivo, Madagascar
rabeherimanana.toussaint@yahoo.fr
10.22124/jart.2018.10977.1110
First, 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.
quotient-ring,ideal,ideal basis,multicyclic code,polynomial division algorithm (by many divisors)
https://jart.guilan.ac.ir/article_3329.html
https://jart.guilan.ac.ir/article_3329_8d8f5e7ba61fd23caa4b4d6f970ceb64.pdf
University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
Quasi-bigraduations of Modules, criteria of generalized analytic independence
79
96
EN
Y. M.
Diagana
0000-0002-7226-6364
Laboratoire Math$acute{e}$matiques-Informatique, Universit$acute{e}$ Nangui Abrogoua, Abidjan, C$hat{o}$te d'Ivoire
y_diagana@yahoo.com
10.22124/jart.2018.11137.1113
Let $\mathcal{R}$ 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.
Quasi-bigraduations,modules,generalized analytic independence
https://jart.guilan.ac.ir/article_3330.html
https://jart.guilan.ac.ir/article_3330_ee9644cecd6586d3400f22645cd4623f.pdf
University of Guilan
Journal of Algebra and Related Topics
2345-3931
2382-9877
6
2
2018
12
01
Prime extension dimension of a module
97
106
EN
T.
Duraivel
Department of Mathematics, Pondicherry University, Puducherry, India.
tduraivel@gmail.com
S.
Mangayarcarassy
Department of Mathematics, Pondicherry Engineering College, Puducherry, India.
dmangay@pec.edu
K.
Premkumar
Department of Mathematics, Indira Gandhi Institute of Technology, Odisha, India.
prem.pondiuni@gmail.com
10.22124/jart.2018.11232.1116
We 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 \(M\), \(\mr{pe.d}_A(N)\leq\mr{pe.d}_A(M)\) and \(\mr{pe.d}_A(N)+\mr{pe.d}_A(M/N)\geq\mr{pe.d}_A(M)\). 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 \(M\).
Prime submodules,Primary decomposition,Prime filtration and Regular prime extension filtration
https://jart.guilan.ac.ir/article_3331.html
https://jart.guilan.ac.ir/article_3331_8a5342e66d28b4c73dccff01968afa06.pdf