eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
79
89
10.22044/jas.2014.228
228
SOME RESULTS ON STRONGLY PRIME SUBMODULES
Alireza Naghipour
naghipourar@yahoo.com
1
Shahrekord University,
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.
http://jas.shahroodut.ac.ir/article_228_6566623d100f92ad63091efa325975a1.pdf
Prime submodule
classical Krull dimension
strongly prime submodule
eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
91
100
10.22044/jas.2014.229
229
A NEW PROOF OF THE PERSISTENCE PROPERTY FOR
IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
Mehrdad Nasernejad
m_nasernejad@yahoo.com
1
member of Iranian Mathematical society, Payeme Noor phd student
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.
http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdf
Dedekind rings
Pr¨ufer domains
weakly associated prime ideals
associated
prime ideals
powers of ideals
eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
101
115
10.22044/jas.2014.230
230
ZARISKI-LIKE SPACES OF CERTAIN MODULES
Hosein Fazaeli Moghim
hfazaeli@birjand.ac.ir
1
Fatemeh Rashedi
fatemehrashedi@birjand.ac.ir
2
Academic member-University of Birjand
Department of Mathematics-University of Birjand
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.
http://jas.shahroodut.ac.ir/article_230_b7b37843a5fe23f4743e67cb83ccec30.pdf
RSP module
Multiplication module
Zariski-like space
Subtractive subsemi-
module
Subtractive basis
eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
117
133
10.22044/jas.2014.231
231
Classification of Lie Subalgebras up to an Inner Automorphism
Seyed Reza Hejazi
ra.hejazi@gmail.com
1
University of Shahrood
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.
http://jas.shahroodut.ac.ir/article_231_7c2bfe95b378521e2f2c00a52d821f78.pdf
Lie algebra
vector fields
optimal system
eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
135
147
10.22044/jas.2014.232
232
Lattice of weak hyper K-ideals of a hyper K-algebra
Mahmood Bakhshi
bakhshi@ub.ac.ir
1
teacher.bojnoord university.iran
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.
http://jas.shahroodut.ac.ir/article_232_fe64e3c27374bc5dcee6428ef6fbdbec.pdf
Hyper K-ideals
weak hyper K-ideals
Boolean lattice
eng
Shahrood University of Technology
Journal of Algebraic Systems
2345-5128
2345-511X
2014-01-01
1
2
149
160
10.22044/jas.2014.233
233
Quasi-Primary Decomposition in Modules Over Proufer Domains
Mahmood Behboodi
mbehbood@cc.iut.ac.ir
1
Reza Jahani-Nezhad
jahanian@kashanu.ac.ir
2
Mohammad Hasan Naderi
mh-naderi@qom.ac.ir
3
Department of Mathematics, Faculty of Science, University of Kashan
Department of Mathematics, Faculty of Science, University of Qom
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.
http://jas.shahroodut.ac.ir/article_233_7d82024d729effde8d1807391f2bc9e3.pdf
Proufer domain
primary submodule
quasi-primary submodule
classical
quasi-primary
Decomposition