Line graphs associated to the maximal graph
A.
Sharma
University of Delhi
author
A.
Gaur
University of Delhi
author
text
article
2015
eng
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 non-unit 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.
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
1
11
http://jart.guilan.ac.ir/article_1209_6febd2a7a22b03870dcd02ddde00b032.pdf
Strongly cotop modules
H.
Ansari-Toroghy
University of Guilan
author
S.S.
Pourmortazavi
University of Guilan
author
S.
Keyvani
Islamic Azad University
author
text
article
2015
eng
In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
13
29
http://jart.guilan.ac.ir/article_1210_4cce280ab88cc7218f63973c940d1f25.pdf
On the fitting ideals of a comultiplication module
S.
Karimzadeh
Vali-e-Asr University of Rafsanjan
author
S.
Hadjirezaei
Vali-e-Asr University of Rafsanjan
author
text
article
2015
eng
Let $R$ be a commutative ring. In this paper we assert some properties of finitely generated comultiplication modules and Fitting ideals of them.
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
31
39
http://jart.guilan.ac.ir/article_1211_33e4f18032e4525d4779c336be03ffab.pdf
F-regularity relative to modules
F.
Dorostkar
University of Guilan
author
R.
khosravi
University of Guilan
author
text
article
2015
eng
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 .
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
41
50
http://jart.guilan.ac.ir/article_1212_1707024d6a7b82e8d1892f7dbb86f9ff.pdf
A note on maximal non-prime ideals
S.
Visweswaran
Saurashtra University
author
A.
Parmar
Saurashtra University
author
text
article
2015
eng
The rings considered in this article are commutative with identity $1\neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $I\neq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ I\subseteq A$ and $I\neq A$ 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 non-maximal ideal and maximal non-primary 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 non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
51
61
http://jart.guilan.ac.ir/article_1213_ef7c4b9f1125da4eb7d276b1d9cbcb6d.pdf
Some numerical results on two classes of finite groups
M.
Hashemi
University of Guilan
author
M.
Polkouei
University of Guilan
author
text
article
2015
eng
In this paper, we consider the finitely presented groups $G_{m}$ and $K(s,l)$ as follows;$$G_{m}=\langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]\rangle $$$$K(s,l)=\langle a,b|ab^s=b^la,~ba^s=a^lb\rangle;$$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, $m,n\geq 2$ and $g.c.d(s,l)=1$.
Journal of Algebra and Related Topics
University of Guilan
2345-3931
3
v.
1
no.
2015
63
72
http://jart.guilan.ac.ir/article_1214_feb823612fb517d4d22669f4bdc86f76.pdf