2014
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2
2
82
SOME RESULTS ON STRONGLY PRIME SUBMODULES
2
2
Let $R$ be a commutative ring with identity and let $M$ be an $R$module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)y P$ for $x, y M$, implies that $x P$ or $y P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$module $M$ is Artinian if and only if $M$ is Noetherian and every strongly prime submodule of $M$ is maximal. We also study the strongly dimension of a module which is defined to be the length of a longest chain of strongly prime submodules.
1

79
89


A.R.
Naghipour
Department of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord,
Iran.
Department of Mathematics, Shahrekord University,
Iran
naghipourar@yahoo.com
Prime submodule
classical Krull dimension
strongly prime submodule
A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
2
2
In this paper, by using elementary tools of commutative algebra, we prove the persistence property for two especial classes of rings. In fact, this paper has two main sections. In the first main section, we let $R$ be a Dedekind ring and $I$ be a proper ideal of $R$. We prove that if $I_1,ldots,I_n$ are nonzero proper ideals of $R$, then ${Ass}^{infty}(I_1^{k_1}ldots I_n^{k_n})={Ass}^{infty}(I_1^{k_1})cupcdotscup {Ass}^{infty}(I_n^{k_n})$ for all $k_1,ldots,k_n geq 1$, where for an ideal $J$ of $R$, ${Ass}^{infty}(J)$ is the stable set of associated primes of $J$. Moreover, we prove that every nonzero ideal in a Dedekind ring is RatliffRush closed, normally torsionfree and also has a strongly superficial element. Especially, we show that if $mathcal{R}=mathcal{R}(R, I)$ is the Rees ring of $R$ with respect to $I$, as a subring of $R[t,u]$ with $u=t^{1}$, then $umathcal{R}$ has no irrelevant prime divisor. par In the second main section, we prove that every nonzero finitely generated ideal in a Pr"{u}fer domain has the persistence property with respect to weakly associated prime ideals.
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91
100


M.
Nasernejad
Department of Mathematics, Payame Noor University, P.O.Box 193953697, Tehran,
Iran.
Department of Mathematics, Payame Noor University,
Iran
m_nasernejad@yahoo.com
Dedekind rings
Pr¨ufer domains
weakly associated prime ideals
associated prime ideals
powers of ideals
ZARISKILIKE SPACES OF CERTAIN MODULES
2
2
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$module. The primarylike spectrum $Spec_L(M)$ is the collection of all primarylike submodules $Q$ such that $M/Q$ is a primeful $R$module. Here, $M$ is defined to be RSP if $rad(Q)$ is a prime submodule for all $Qin Spec_L(M)$. This class contains the family of multiplication modules properly. The purpose of this paper is to introduce and investigate a new Zariski space of an RSP module, called Zariskilike space. In particular, we provide conditions under which the Zariskilike space of a multiplication module has a subtractive basis.
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101
115


H.
Fazaeli Moghim
Department of Mathematics, Department of Mathematics, University of Birjand,
P.O. Box 97175615, Birjand, Iran.
Department of Mathematics, Department of
Iran
hfazaeli@birjand.ac.ir


F.
Rashedi
Department of Mathematics, University of Birjand, P.O. Box 97175615, Birjand,
Iran.
Department of Mathematics, University of
Iran
fatemehrashedi@birjand.ac.ir
RSP module
Multiplication module
Zariskilike space
Subtractive subsemi module
Subtractive basis
CLASSIFICATION OF LIE SUBALGEBRAS UP TO AN INNER AUTOMORPHISM
2
2
In this paper, a useful classification of all Lie subalgebras of a given Lie algebra up to an inner automorphism is presented. This method can be regarded as an important connection between differential geometry and algebra and has many applications in different fields of mathematics. After main results, we have applied this procedure for classifying the Lie subalgebras of some examples of Lie algebras.
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117
133


Seyed R.
Hejazi
Department of Mathematics, Shahrood University, Shahrood, IRAN.
Department of Mathematics, Shahrood University,
Iran
ra.hejazi@gmail.com
Lie algebra
vector fields
optimal system
Lattice of weak hyper Kideals of a hyper Kalgebra
2
2
In this note, we study the lattice structure on the class of all weak hyper Kideals of a hyper Kalgebra. We first introduce the notion of (left,right) scalar in a hyper Kalgebra which help us to characterize the weak hyper Kideals generated by a subset. In the sequel, using the notion of a closure operator, we study the lattice of all weak hyper Kideals of a hyper Kalgebra, and we prove a special subclass of this class together with the suitable operations forms a Boolean lattice.
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135
147


M.
Bakhshi
Department of Mathematics, University of Bojnord, P.O.Box 1339, Bojnord, Iran.
Department of Mathematics, University of
Iran
bakhshi@ub.ac.ir
Hyper Kideals
weak hyper Kideals
Boolean lattice
QuasiPrimary Decomposition in Modules Over Proufer Domains
2
2
In this paper we investigate decompositions of submodules in modules over a Proufer domain into intersections of quasiprimary and classical quasiprimary submodules. In particular, existence and uniqueness of quasiprimary decompositions in modules over a Proufer domain of ﬁnite character are proved.
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149
160


M.
Behboodi
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box
8415683111, Isfahan, Iran, and
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 193955746, Tehran, Iran.
Department of Mathematical Sciences, Isfahan
Iran
mbehbood@cc.iut.ac.ir


R.
JahaniNezhad
Department of Mathematics, Faculty of Science, University of Kashan, Kashan,
Iran.
Department of Mathematics, Faculty of Science,
Iran
jahanian@kashanu.ac.ir


M. H.
Naderi
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran.
Department of Mathematics, Faculty of Science,
Iran
mhnaderi@qom.ac.ir
Proufer domain
primary submodule
quasiprimary submodule
classical quasiprimary
Decomposition