SOME RESULTS ON STRONGLY PRIME SUBMODULES
Alireza
Naghipour
Shahrekord University,
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
79
89
http://jas.shahroodut.ac.ir/article_228_6566623d100f92ad63091efa325975a1.pdf
dx.doi.org/10.22044/jas.2014.228
A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
Mehrdad
Nasernejad
member of Iranian Mathematical society, Payeme Noor phd student
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
91
100
http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdf
dx.doi.org/10.22044/jas.2014.229
ZARISKI-LIKE SPACES OF CERTAIN MODULES
Hosein
Fazaeli Moghim
Academic member-University of Birjand
author
Fatemeh
Rashedi
Department of Mathematics-University of Birjand
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
101
115
http://jas.shahroodut.ac.ir/article_230_b7b37843a5fe23f4743e67cb83ccec30.pdf
dx.doi.org/10.22044/jas.2014.230
Classification of Lie Subalgebras up to an Inner Automorphism
Seyed Reza
Hejazi
University of Shahrood
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
117
133
http://jas.shahroodut.ac.ir/article_231_7c2bfe95b378521e2f2c00a52d821f78.pdf
dx.doi.org/10.22044/jas.2014.231
Lattice of weak hyper K-ideals of a hyper K-algebra
Mahmood
Bakhshi
teacher.bojnoord university.iran
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
135
147
http://jas.shahroodut.ac.ir/article_232_fe64e3c27374bc5dcee6428ef6fbdbec.pdf
dx.doi.org/10.22044/jas.2014.232
Quasi-Primary Decomposition in Modules Over Proufer Domains
Mahmood
Behboodi
author
Reza
Jahani-Nezhad
Department of Mathematics, Faculty of Science, University of Kashan
author
Mohammad Hasan
Naderi
Department of Mathematics, Faculty of Science, University of Qom
author
text
article
2014
eng
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.
Journal of Algebraic Systems
Shahrood University of Technology
2345-5128
1
v.
2
no.
2014
149
160
http://jas.shahroodut.ac.ir/article_233_7d82024d729effde8d1807391f2bc9e3.pdf
dx.doi.org/10.22044/jas.2014.233