ORIGINAL_ARTICLE
SOME RESULTS ON STRONGLY PRIME SUBMODULES
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.
http://jas.shahroodut.ac.ir/article_228_6566623d100f92ad63091efa325975a1.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
79
89
10.22044/jas.2014.228
Prime submodule
classical Krull dimension
strongly prime submodule
A.R.
Naghipour
naghipourar@yahoo.com
true
1
Department of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord,
Iran.
Department of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord,
Iran.
Department of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord,
Iran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS
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 non-zero proper ideals of $R$, then ${Ass}^{\infty}(I_1^{k_1}\ldots I_n^{k_n})={Ass}^{\infty}(I_1^{k_1})\cup\cdots\cup {Ass}^{\infty}(I_n^{k_n})$ 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 non-zero ideal in a Dedekind ring is Ratliff-Rush closed, normally torsion-free 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 $u\mathcal{R}$ has no irrelevant prime divisor. \par In the second main section, we prove that every non-zero finitely generated ideal in a Pr\"{u}fer domain has the persistence property with respect to weakly associated prime ideals.
http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
91
100
10.22044/jas.2014.229
Dedekind rings
Pr¨ufer domains
weakly associated prime ideals
associated prime ideals
powers of ideals
M.
Nasernejad
m_nasernejad@yahoo.com
true
1
Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran,
Iran.
Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran,
Iran.
Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran,
Iran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
ZARISKI-LIKE SPACES OF CERTAIN MODULES
Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. The primary-like spectrum $Spec_L(M)$ is the collection of all primary-like 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 $Q\in Spec_L(M)$. 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 Zariski-like space. In particular, we provide conditions under which the Zariski-like space of a multiplication module has a subtractive basis.
http://jas.shahroodut.ac.ir/article_230_b7b37843a5fe23f4743e67cb83ccec30.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
101
115
10.22044/jas.2014.230
RSP module
Multiplication module
Zariski-like space
Subtractive subsemi- module
Subtractive basis
H.
Fazaeli Moghim
hfazaeli@birjand.ac.ir
true
1
Department of Mathematics, Department of Mathematics, University of Birjand,
P.O. Box 97175-615, Birjand, Iran.
Department of Mathematics, Department of Mathematics, University of Birjand,
P.O. Box 97175-615, Birjand, Iran.
Department of Mathematics, Department of Mathematics, University of Birjand,
P.O. Box 97175-615, Birjand, Iran.
LEAD_AUTHOR
F.
Rashedi
fatemehrashedi@birjand.ac.ir
true
2
Department of Mathematics, University of Birjand, P.O. Box 97175-615, Birjand,
Iran.
Department of Mathematics, University of Birjand, P.O. Box 97175-615, Birjand,
Iran.
Department of Mathematics, University of Birjand, P.O. Box 97175-615, Birjand,
Iran.
AUTHOR
ORIGINAL_ARTICLE
CLASSIFICATION OF LIE SUBALGEBRAS UP TO AN INNER AUTOMORPHISM
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.
http://jas.shahroodut.ac.ir/article_231_7c2bfe95b378521e2f2c00a52d821f78.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
117
133
10.22044/jas.2014.231
Lie algebra
vector fields
optimal system
Seyed R.
Hejazi
ra.hejazi@gmail.com
true
1
Department of Mathematics, Shahrood University, Shahrood, IRAN.
Department of Mathematics, Shahrood University, Shahrood, IRAN.
Department of Mathematics, Shahrood University, Shahrood, IRAN.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Lattice of weak hyper K-ideals of a hyper K-algebra
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 a hyper K-algebra, and we prove a special subclass of this class together with the suitable operations forms a Boolean lattice.
http://jas.shahroodut.ac.ir/article_232_fe64e3c27374bc5dcee6428ef6fbdbec.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
135
147
10.22044/jas.2014.232
Hyper K-ideals
weak hyper K-ideals
Boolean lattice
M.
Bakhshi
bakhshi@ub.ac.ir
true
1
Department of Mathematics, University of Bojnord, P.O.Box 1339, Bojnord, Iran.
Department of Mathematics, University of Bojnord, P.O.Box 1339, Bojnord, Iran.
Department of Mathematics, University of Bojnord, P.O.Box 1339, Bojnord, Iran.
LEAD_AUTHOR
ORIGINAL_ARTICLE
Quasi-Primary Decomposition in Modules Over Proufer Domains
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.
http://jas.shahroodut.ac.ir/article_233_7d82024d729effde8d1807391f2bc9e3.pdf
2014-01-01T11:23:20
2020-06-07T11:23:20
149
160
10.22044/jas.2014.233
Proufer domain
primary submodule
quasi-primary submodule
classical quasi-primary
Decomposition
M.
Behboodi
mbehbood@cc.iut.ac.ir
true
1
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box
84156-83111, Isfahan, Iran, and
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5746, Tehran, Iran.
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box
84156-83111, Isfahan, Iran, and
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5746, Tehran, Iran.
Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box
84156-83111, Isfahan, Iran, and
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5746, Tehran, Iran.
LEAD_AUTHOR
R.
Jahani-Nezhad
jahanian@kashanu.ac.ir
true
2
Department of Mathematics, Faculty of Science, University of Kashan, Kashan,
Iran.
Department of Mathematics, Faculty of Science, University of Kashan, Kashan,
Iran.
Department of Mathematics, Faculty of Science, University of Kashan, Kashan,
Iran.
AUTHOR
M. H.
Naderi
mh-naderi@qom.ac.ir
true
3
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran.
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran.
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran.
AUTHOR