Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101SOME RESULTS ON STRONGLY PRIME SUBMODULES798922810.22044/jas.2014.228ENA.R.NaghipourDepartment of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord,
Iran.0000-0002-7178-6173Journal Article20130411Let $R$ 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 <br />which is defined to be the length of a longest chain of strongly prime submodules.Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101A NEW PROOF OF THE PERSISTENCE PROPERTY FOR IDEALS IN DEDEKIND RINGS AND PR¨UFER DOMAINS9110022910.22044/jas.2014.229ENM.NasernejadDepartment of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran,
Iran.Journal Article20130503In 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 ${Ass}^{infty}(I_1^{k_1}ldots I_n^{k_n})={Ass}^{infty}(I_1^{k_1})cupcdotscup {Ass}^{infty}(I_n^{k_n})$ 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 $umathcal{R}$ 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.Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101ZARISKI-LIKE SPACES OF CERTAIN MODULES10111523010.22044/jas.2014.230ENH.Fazaeli MoghimDepartment of Mathematics, Department of Mathematics, University of Birjand,
P.O. Box 97175-615, Birjand, Iran.F.RashediDepartment of Mathematics, University of Birjand, P.O. Box 97175-615, Birjand,
Iran.Journal Article20130413Let $R$ 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 $Qin Spec_L(M)$. 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.Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101CLASSIFICATION OF LIE SUBALGEBRAS UP TO AN INNER AUTOMORPHISM11713323110.22044/jas.2014.231ENSeyed R.HejaziDepartment of Mathematics, Shahrood University, Shahrood, IRAN.Journal Article20130606In 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.Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101Lattice of weak hyper K-ideals of a hyper K-algebra13514723210.22044/jas.2014.232ENM.BakhshiDepartment of Mathematics, University of Bojnord, P.O.Box 1339, Bojnord, Iran.0000-0001-6552-1307Journal 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 hyper K-algebra, and we prove a special subclass of this class together with the suitable operations forms a Boolean lattice.Shahrood University of TechnologyJournal of Algebraic Systems2345-51281220140101Quasi-Primary Decomposition in Modules Over Proufer Domains14916023310.22044/jas.2014.233ENM.BehboodiDepartment 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.R.Jahani-NezhadDepartment of Mathematics, Faculty of Science, University of Kashan, Kashan,
Iran.M. H.NaderiDepartment of Mathematics, Faculty of Science, University of Qom, Qom, Iran.Journal Article20130314In 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.