Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101SOME RESULTS ON STRONGLY PRIME SUBMODULES798922810.22044/jas.2014.228ENAlireza NaghipourShahrekord University,Journal Article20130411Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. <br />A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : <br />M)ysubseteq P$ 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 <br />$R$-module $M$ is Artinian if and only if $M$ is Noetherian and every strongly prime <br />submodule of $M$ is maximal. We also study the strongly dimension of a module <br />which is defined to be the length of a longest chain of strongly prime submodules.http://jas.shahroodut.ac.ir/article_228_6566623d100f92ad63091efa325975a1.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101A NEW PROOF OF THE PERSISTENCE PROPERTY FOR
IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS9110022910.22044/jas.2014.229ENMehrdad Nasernejadmember of Iranian Mathematical society, Payeme Noor phd studentJournal Article20130503In this paper, by using elementary tools of commutative algebra,<br />we prove the persistence property for two especial classes of rings. In fact, this<br />paper has two main sections. In the first main section, we let R be a Dedekind<br />ring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zero<br />proper ideals of R, then Ass1(Ik1<br />1 . . . Ikn<br />n ) = Ass1(Ik1<br />1 ) [ · · · [ Ass1(Ikn<br />n )<br />for all k1, . . . , kn 1, where for an ideal J of R, Ass1(J) is the stable set<br />of associated primes of J. Moreover, we prove that every non-zero ideal in<br />a Dedekind ring is Ratliff-Rush closed, normally torsion-free and also has a<br />strongly superficial element. Especially, we show that if R = R(R, I) is the<br />Rees ring of R with respect to I, as a subring of R[t, u] with u = t−1, then uR<br />has no irrelevant prime divisor.<br />In the second main section, we prove that every non-zero finitely generated<br />ideal in a Pr¨ufer domain has the persistence property with respect to weakly<br />associated prime ideals. Finally, we extend the notion of persistence property<br />of ideals to the persistence property for rings.http://jas.shahroodut.ac.ir/article_229_aedeac2f9e82c3042ad040a8f3f9241a.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101ZARISKI-LIKE SPACES OF CERTAIN MODULES10111523010.22044/jas.2014.230ENHosein Fazaeli MoghimAcademic member-University of BirjandFatemeh RashediDepartment of Mathematics-University of BirjandJournal Article20130413Let $R$ be a commutative ring with identity and $M$ be a unitary<br />$R$-module. The primary-like spectrum $Spec_L(M)$ is the<br />collection of all primary-like submodules $Q$ such that $M/Q$ is a<br />primeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ is<br />a prime submodule for all $Qin Spec_L(M)$. This class contains<br />the family of multiplication modules properly. The purpose of this<br />paper is to introduce and investigate a new Zariski space of an<br />RSP module, called Zariski-like space. In particular, we provide<br />conditions under which the Zariski-like space of a multiplication<br />module has a subtractive basis.http://jas.shahroodut.ac.ir/article_230_b7b37843a5fe23f4743e67cb83ccec30.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101Classification of Lie Subalgebras up to an Inner Automorphism11713323110.22044/jas.2014.231ENSeyed Reza HejaziUniversity of ShahroodJournal Article20130606In this paper, a useful classification of all Lie subalgebras of a given Lie algebra<br />up to an inner automorphism is presented. This method can be regarded as an<br />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.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101Lattice of weak hyper K-ideals of a hyper K-algebra13514723210.22044/jas.2014.232ENMahmood Bakhshiteacher.bojnoord university.iranJournal Article20130903In 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<br />hyper K-algebra, and we prove a special subclass of this class together<br />with the suitable operations forms a Boolean lattice.http://jas.shahroodut.ac.ir/article_232_fe64e3c27374bc5dcee6428ef6fbdbec.pdfShahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101Quasi-Primary Decomposition in Modules Over Proufer Domains14916023310.22044/jas.2014.233ENMahmood BehboodiReza Jahani-NezhadDepartment of Mathematics, Faculty of Science, University of KashanMohammad Hasan NaderiDepartment of Mathematics, Faculty of Science, University of QomJournal Article20130314In this paper we investigate decompositions of submodules in modules over a Proufer <br />domain into intersections of quasi-primary and classical quasi-primary submodules. <br />In particular, existence and uniqueness of quasi-primary decompositions in modules <br />over a Proufer domain of ﬁnite character are proved. <br /> <br /> <br />Proufer domain; primary submodule; quasi-primary submodule; classical <br />quasi-primary; decomposition.http://jas.shahroodut.ac.ir/article_233_7d82024d729effde8d1807391f2bc9e3.pdf