2014
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2
2
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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 $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. 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


Alireza
Naghipour
Shahrekord University,
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, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are nonzeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1, . . . , kn 1, where for an ideal J of R, Ass1(J) is the stable setof associated primes of J. Moreover, we prove that every nonzero ideal ina Dedekind ring is RatliffRush closed, normally torsionfree and also has astrongly superficial element. Especially, we show that if R = R(R, I) is theRees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uRhas no irrelevant prime divisor.In the second main section, we prove that every nonzero finitely generatedideal in a Pr¨ufer domain has the persistence property with respect to weaklyassociated prime ideals. Finally, we extend the notion of persistence propertyof ideals to the persistence property for rings.
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91
100


Mehrdad
Nasernejad
member of Iranian Mathematical society, Payeme Noor phd student
member of Iranian Mathematical society, Payeme
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 thecollection of all primarylike submodules $Q$ such that $M/Q$ is aprimeful $R$module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to introduce and investigate a new Zariski space of anRSP module, called Zariskilike space. In particular, we provideconditions under which the Zariskilike space of a multiplicationmodule has a subtractive basis.
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101
115


Hosein
Fazaeli Moghim
Academic memberUniversity of Birjand
Academic memberUniversity of Birjand
Iran
hfazaeli@birjand.ac.ir


Fatemeh
Rashedi
Department of MathematicsUniversity of Birjand
Department of MathematicsUniversity of Birjand
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 algebraup to an inner automorphism is presented. This method can be regarded as animportant 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.
1

117
133


Seyed Reza
Hejazi
University of Shahrood
University of Shahrood
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 ahyper Kalgebra, and we prove a special subclass of this class togetherwith the suitable operations forms a Boolean lattice.
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135
147


Mahmood
Bakhshi
teacher.bojnoord university.iran
teacher.bojnoord university.iran
Iran
bakhshi@ub.ac.ir
Hyper Kideals
weak hyper Kideals
Boolean lattice
QuasiPrimary Decomposition in Modules Over Proufer Domains
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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. Proufer domain; primary submodule; quasiprimary submodule; classical quasiprimary; decomposition.
1

149
160


Mahmood
Behboodi
Iran
mbehbood@cc.iut.ac.ir


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


Mohammad Hasan
Naderi
Department of Mathematics, Faculty of Science, University of Qom
Department of Mathematics, Faculty of Science,
Iran
mhnaderi@qom.ac.ir
Proufer domain
primary submodule
quasiprimary submodule
classical quasiprimary
decomposition