2018
5
2
0
86
A COVERING PROPERTY IN PRINCIPAL BUNDLES
2
2
Let $p:Xlo B$ be a locally trivial principal Gbundle and $wt{p}:wt{X}lo B$ be a locally trivial principal $wt{G}$bundle. In this paper, by using the structure of principal bundles according to transition functions, we show that $wt{G}$ is a covering group of $G$ if and only if $wt{X}$ is a covering space of $X$. Then we conclude that a topological space $X$ with nonsimply connected universal covering space has no connected locally trivial principal $pi(X,x_0)$bundle, for every $x_0in X$.
1

91
98


A.
Pakdaman
Department of Mathematics, University of Golestan, P.O.Box 155, Gorgan, Iran.
Department of Mathematics, University of
Iran
a.pakdaman@gu.ac.ir


M.
Attary
Department of Mathematics, University of Golestan, P.O.Box 155, Gorgan, Iran.
Department of Mathematics, University of
Iran
m_atari1989@yahoo.com
Principal bundle
covering space
covering group
ON (n 1; n)phimPRIME AND (n 1; n)WEAKLY PRIME SUBMODULES
2
2
Abstract. Let m; n ≥ 2 be two positive integers, R a commutative ring with identity and M a unitary Rmodule. A proper submodule P of M is an (n 1; n)Φmprime ((n 1; n)weakly prime) submodule if a1; : : : ; an1 2 R and x 2 M together with a1 : : : an1x 2 Pn(P : M)m1P (0 ̸= a1 : : : an1x 2 P) imply a1 : : : ai1ai+1 : : : an1x 2 P, for some i 2 f1; : : : ; n1g or a1:::an1 2 (P : M). In this paper we study these submodules. Some useful results and examples concerning these types of submodules are given.
1

99
109


M.
Ebrahimpour
Department of Mathematics, Faculty of Sciences, ValieAsr University of Rafsanjan
, P.O.Box 518, Rafsanjan, Iran
Department of Mathematics, Faculty of Sciences,
Iran
m.ebrahimpour@vru.ac.ir


F.
Mirzaee
Department of Mathematics, Faculty of Sciences, Shahid Bahonar University of
Kerman, Kerman, Iran.
Department of Mathematics, Faculty of Sciences,
Iran
mirzaee0269@yahoo.com
Quasilocal ring, Weakly prime submodule, (n1
n)weakly prime submodule, ϕmprime submodule
SEQUENTIALLY COMPACT SACTS
2
2
The investigation of equational compactness was initiated by Banaschewski and Nelson. They proved that pure injectivity is equivalent to equational compactness. Here we define the so called sequentially compact acts over semigroups and study some of their categorical and homological properties. Some Baer conditions for injectivity of Sacts are also presented.
1

111
125


H.
Barzegar
Department of Mathematics, University of Tafresh , 3951879611, Tafresh, Iran.
Department of Mathematics, University of
Iran
h56bar@tafreshu.ac.ir
sequentially compact
$f$pure injective
injective Sact
TOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS
2
2
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by pointattaching from $G_1, ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. In this paper, we consider some particular cases of these graphs that most of them are of importance in chemistry and study their total domination polynomials.
1

127
138


S.
Alikhani
Department of Mathematics, Yazd University, 89195741, Yazd, Iran.
Department of Mathematics, Yazd University,
Iran
alikhani@yazd.ac.ir


N.
Jafari
Department of Mathematics, Yazd University, 89195741 Yazd, Iran.
Department of Mathematics, Yazd University,
Iran
nasrin7190@yahoo.com
Total domination number
total domination polynomial
total dominating set
ON pNILPOTENCY OF FINITE GROUPS WITH SSNORMAL SUBGROUPS
2
2
Abstract. A subgroup H of a group G is said to be SSembeddedin G if there exists a normal subgroup T of G such that HT issubnormal in G and H T ≤ H sG , where H sG is the maximal spermutable subgroup of G contained in H. We say that a subgroupH is an SSnormal subgroup in G if there exists a normal subgroupT of G such that G = HT and H T ≤ H SS , where H SS is anSSembedded subgroup of G contained in H. In this paper, westudy the inﬂuence of some SSnormal subgroups on the structureof a ﬁnite group G.
1

139
148


G. R.
REZAEEZADEH
Department of Mathematics, University of Shahrekord, P.O.Box 115, Shahrekord,
Iran.
Department of Mathematics, University of
Iran
gh.rezaeezadeh@yahoo.com


Z.
AGHAJARI
Department of Mathematics, University of Shahrekord, P.O.Box 115, Shahrekord,
Iran.
Department of Mathematics, University of
Iran
z.aghajari@stu.sku.ac.ir
SSnormal subgroup
SSembedded subgroup
pnilpotent group
INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REALVALUED CONTINUOUS FUNCTIONS ON A FRAME
2
2
A frame $L$ is called {it cozdense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of realvalued continuous functions on a cozdense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essential ideal in $mathcal RL$ if and only if the set of isolated points of $ Sigma L$ is dense in $ Sigma L$ if and only if the intersection of any family of essential ideals is essential in $mathcal RL$. Besides, the counterpart of some results in the ring $C(X)$ is studied for the ring $mathcal RL$. For example, an ideal $E$ of $mathcal RL$ is an essential ideal if and only if $bigcap Z[E]$ is a nowhere dense subset of $Sigma L.$
1

149
161


A. A.
Estaji
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze
var, Iran.
Faculty of Mathematics and Computer Sciences,
Iran
aaestaji@hsu.ac.ir


A. Gh.
Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan,
Department of Mathematics, Gorgan Branch,
Iran
akarimi@gorganiau.ac.ir


M.
Abedi
Esfarayen University of Technology, Esfarayen, Iran.
Esfarayen University of Technology, Esfarayen,
Iran
abedi@esfarayen.ac.ir
Frame
essential ideal
socle
zero sets in pointfree topology
ring of realvalued continuous functions on a frame
A GENERALIZATION OF CORETRACTABLE MODULES
2
2
Let $R$ be a ring and $M$ a right $R$module. We call $M$, coretractable relative to $overline{Z}(M)$ (for short, $overline{Z}(M)$coretractable) provided that, for every proper submodule $N$ of $M$ containing $overline{Z}(M)$, there is a nonzero homomorphism $f:dfrac{M}{N}rightarrow M$. We investigate some conditions under which the two concepts coretractable and $overline{Z}(M)$coretractable, coincide. For a commutative semiperfect ring $R$, we show that $R$ is $overline{Z}(R)$coretractable if and only if $R$ is a Kasch ring. Some examples are provided to illustrate different concepts.
1

163
176


A. R.
Moniri Hamzekolaee
Department of Mathematics, University of Mazandaran, Babolsar, Iran
Department of Mathematics, University of
Iran
a.monirih@umz.ac.ir
coretractable module
$overline{Z}(M)$coretractable module
Kasch ring