Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
SOME RESULTS ON STRONGLY PRIME SUBMODULES
79
89
EN
Alireza
Naghipour
Shahrekord University,
naghipourar@yahoo.com
10.22044/jas.2014.228
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.
Prime submodule,classical Krull dimension,strongly prime submodule
http://jas.shahroodut.ac.ir/article_228.html
http://jas.shahroodut.ac.ir/article_228_6566623d100f92ad63091efa325975a1.pdf
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
91
100
EN
Mehrdad
Nasernejad
member of Iranian Mathematical society, Payeme Noor phd student
m_nasernejad@yahoo.com
10.22044/jas.2014.229
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 non-zeroproper 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 non-zero ideal ina Dedekind ring is Ratliff-Rush closed, normally torsion-free 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 non-zero 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.
Dedekind rings,Pr¨ufer domains,weakly associated prime ideals,associated prime ideals,powers of ideals
http://jas.shahroodut.ac.ir/article_229.html
http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdf
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
ZARISKI-LIKE SPACES OF CERTAIN MODULES
101
115
EN
Hosein
Fazaeli Moghim
Academic member-University of Birjand
hfazaeli@birjand.ac.ir
Fatemeh
Rashedi
Department of Mathematics-University of Birjand
fatemehrashedi@birjand.ac.ir
10.22044/jas.2014.230
Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like 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 Zariski-like space. In particular, we provideconditions under which the Zariski-like space of a multiplicationmodule has a subtractive basis.
RSP module,Multiplication module,Zariski-like space,Subtractive subsemi-
module,Subtractive basis
http://jas.shahroodut.ac.ir/article_230.html
http://jas.shahroodut.ac.ir/article_230_b7b37843a5fe23f4743e67cb83ccec30.pdf
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
Classification of Lie Subalgebras up to an Inner Automorphism
117
133
EN
Seyed Reza
Hejazi
University of Shahrood
ra.hejazi@gmail.com
10.22044/jas.2014.231
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.
Lie algebra,vector fields,optimal system
http://jas.shahroodut.ac.ir/article_231.html
http://jas.shahroodut.ac.ir/article_231_7c2bfe95b378521e2f2c00a52d821f78.pdf
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
Lattice of weak hyper K-ideals of a hyper K-algebra
135
147
EN
Mahmood
Bakhshi
teacher.bojnoord university.iran
bakhshi@ub.ac.ir
10.22044/jas.2014.232
In this note, we study the lattice structure on the class of all weak hyper K-ideals of a hyper K-algebra. We first introduce the notion of (left,right) scalar in a hyper K-algebra which help us to characterize the weak hyper K-ideals generated by a subset. In the sequel, using the notion of a closure operator, we study the lattice of all weak hyper K-ideals of ahyper K-algebra, and we prove a special subclass of this class togetherwith the suitable operations forms a Boolean lattice.
Hyper K-ideals,weak hyper K-ideals,Boolean lattice
http://jas.shahroodut.ac.ir/article_232.html
http://jas.shahroodut.ac.ir/article_232_fe64e3c27374bc5dcee6428ef6fbdbec.pdf
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
1
2
2014
01
01
Quasi-Primary Decomposition in Modules Over Proufer Domains
149
160
EN
Mahmood
Behboodi
mbehbood@cc.iut.ac.ir
Reza
Jahani-Nezhad
Department of Mathematics, Faculty of Science, University of Kashan
jahanian@kashanu.ac.ir
Mohammad Hasan
Naderi
Department of Mathematics, Faculty of Science, University of Qom
mh-naderi@qom.ac.ir
10.22044/jas.2014.233
In this paper we investigate decompositions of submodules in modules over a Proufer domain into intersections of quasi-primary and classical quasi-primary submodules. In particular, existence and uniqueness of quasi-primary decompositions in modules over a Proufer domain of ﬁnite character are proved. Proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decomposition.
Proufer domain,primary submodule,quasi-primary submodule,classical
quasi-primary,Decomposition
http://jas.shahroodut.ac.ir/article_233.html
http://jas.shahroodut.ac.ir/article_233_7d82024d729effde8d1807391f2bc9e3.pdf