2015
3
1
1
0
Line graphs associated to the maximal graph
2
2
Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to nonunit elements of $R$. In this paper we study the various graphical properties of the line graph associated to $Gamma(R)$, denoted by $(Gamma(R))$ such that diameter, completeness, and Eulerian property. A complete characterization of rings is given for which $diam(L(Gamma(R)))= diam(Gamma(R))$ or $diam(L(Gamma(R)))< diam(Gamma(R))$ or $diam((Gamma(R)))> diam(Gamma(R))$. We have shown that the complement of the maximal graph $G(R)$, i.e., the comaximal graph is a Euler graph if and only if $R$ has odd cardinality. We also discuss the Eulerian property of the line graph associated to the comaximal graph.
1

1
11


A.
Sharma
University of Delhi
University of Delhi
India
anirudh.maths@gmail.com


A.
Gaur
University of Delhi
University of Delhi
India
gaursatul@gmail.com
Maximal graph
line graph
eulerian graph
comaximal graph
Strongly cotop modules
2
2
In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.
1

13
29


H.
AnsariToroghy
University of Guilan
University of Guilan
Iran
ansari@guilan.ac.ir


S.S.
Pourmortazavi
University of Guilan
University of Guilan
Iran
mortazavi@phd.guilan.ac.ir


S.
Keyvani
Islamic Azad University
Islamic Azad University
Iran
keivani@bandaranzaliiau.ac.ir
Second submodule
strongly cotop module
Zariski topology
spectral space
On the fitting ideals of a comultiplication module
2
2
Let $R$ be a commutative ring. In this paper we assert some properties of finitely generated comultiplication modules and Fitting ideals of them.
1

31
39


S.
Karimzadeh
ValieAsr University of Rafsanjan
ValieAsr University of Rafsanjan
Iran
karimzadeh@vru.ac.ir


S.
Hadjirezaei
ValieAsr University of Rafsanjan
ValieAsr University of Rafsanjan
Iran
s.hajirezaei@vru.ac.ir
Fitting ideals
comultiplication module
simple module
Fregularity relative to modules
2
2
In this paper we will generalize some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .
1

41
50


F.
Dorostkar
University of Guilan
University of Guilan
Iran
dorostkar@guilan.ac.ir


R.
khosravi
University of Guilan
University of Guilan
Iran
khosravi@phd.guilan.ac.ir
Tight closure
$F$regular
and weakly $F$regular relative to a module
A note on maximal nonprime ideals
2
2
The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal nonprime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal. The concept of maximal nonmaximal ideal and maximal nonprimary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal nonprime (respectively, a maximal non maximal, a maximal nonprimary) ideal of $R$.
1

51
61


S.
Visweswaran
Saurashtra University
Saurashtra University
India
visweswaran2006@yahoo.co.in


A.
Parmar
Saurashtra University
Saurashtra University
India
anirudh.maths@gmail.com
Maximal nonprime ideal
maximal nonmaximal ideal
maximal nonprimary ideal
maximal nonirreducible ideal
Some numerical results on two classes of finite groups
2
2
In this paper, we consider the finitely presented groups $G_{m}$ and $K(s,l)$ as follows;$$G_{m}=langle a,b a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$K(s,l)=langle a,bab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$commutativity degree for each of them. Also we study the concept of $n$abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,l)=1$.
1

63
72


M.
Hashemi
University of Guilan
University of Guilan
Iran
m_hashemi@guilan.ac.ir


M.
Polkouei
University of Guilan
University of Guilan
Iran
mikhakp@yahoo.com
Nilpotent groups
$n^{th}$commutativity degree
$n$abelian groups