2015
3
2
2
71
A class of Jquasipolar rings
2
2
In this paper, we introduce a class of $J$quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {it weakly $J$quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $ap$ are contained in $J(R)$ and the ring $R$ is called {it weakly $J$quasipolar} if every element of $R$ is weakly $J$quasipolar. We give many characterizations and investigate general properties of weakly $J$quasipolar rings. If $R$ is a weakly $J$quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is reduced. We use weakly $J$quasipolar rings to obtain more results for $J$quasipolar rings. We prove that the class of weakly $J$quasipolar rings lies between the class of $J$quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring $R$ is abelian weakly $J$quasipolar if and only if $R$ is uniquely clean.
1

1
15


S.
Halicioglu
Ankara University
Ankara University
Turkey
saithalicioglu@gmail.com


M. B.
Calci
Ankara University
Ankara University
Turkey
mburakcalci@gmail.com


A.
Harmanci
Hacettepe University
Hacettepe University
Turkey
harmanci@ankara.edu.tr
Quasipolar ring
$J$quasipolar ring
weakly $J$quasipolar ring
uniquely clean ring
Omegaalmost Boolean rings
2
2
In this paper the concept of an $Omega$ Almost Boolean ring is introduced and illistrated how a sheaf of algebras can be constructed from an $Omega$ Almost Boolean ring over a locally Boolean space.
1

17
25


M. Phani
Krishna Kishore
GVP college of Engineering (Autonomous)
GVP college of Engineering (Autonomous)
India
kishorempk73@gvpce.ac.in
Almost Boolean Rings
Sheaves over locally Boolean spaces
sheaf representations
The total graph of a commutative semiring with respect to proper ideals
2
2
Let $I$ be a proper ideal of a commutative semiring $R$ and let $P(I)$ be the set of all elements of $R$ that are not prime to $I$. In this paper, we investigate the total graph of $R$ with respect to $I$, denoted by $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)$. The properties and possible structures of the two (induced) subgraphs $P(Gamma_{I} (R))$ and $bar {P}(Gamma_{I} (R))$ of $T(Gamma_{I} (R))$, with vertices $P(I)$ and $R  P(I)$, respectively are studied.
1

27
41


Z.
Ebrahimi Sarvandi
University of Guilan
University of Guilan
Iran
zahra_2006_ebrahimi@yahoo.com


S.
Ebrahimi Atani
University of Guilan
University of Guilan
Iran
ebrahimi@guilan.ac.ir
Commutative semirings
Zerodivisor
Total graph
Small submodules with respect to an arbitrary submodule
2
2
Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$module $M$. A submodule $N$ of $M$ is called $T$small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.
1

43
51


R.
Beyranvand
Lorestan University
Lorestan University
Iran
beyranvand.r@lu.ac.ir


F.
Moradi
Lorestan University
Lorestan University
Iran
moradi.fa@fa.lu.ac.ir
Small submodule
Tsmall submodule
Tmaximal submodule
CastelnuovoMumford regularity of products of monomial ideals
2
2
Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l}^{a_l})$.
1

53
59


S.
Yang
Soochow University
Soochow University
China
20144207034@stu.suda.edu.cn


L.
Chu
Soochow University
Soochow University
China
chulizhong@suda.edu.cn


Y.
Qian
Soochow University
Soochow University
China
qianyufenga@gmail.com
CastelnuovoMumford regularity
complete intersections
ideals of Borel type
nthroots and ncentrality of finite 2generator pgroups of nilpotency class 2
2
2
Here we consider all finite nonabelian 2generator $p$groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$roots of them. Also we find integers $n$ for which, these groups are $n$central.
1

61
71


M.
Polkouei
University of Guilan
University of Guilan
Iran
mikhakp@yahoo.com


M.
Hashemi
University of Guilan
University of Guilan
Iran
m_hashemi@guilan.ac.ir
pgroup
nthroots
ncentral group