Journal of Algebraic Systems
Volume:1 Issue: 2, Winter - Spring 2014

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.

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 I1, . . . , In are non-zero proper ideals of R, then Ass1(Ik1 1 . . . Ikn n ) = Ass1(Ik1 1 ) [ · · · [ Ass1(Ikn n ) for all k1, . . . , kn  1, where for an ideal J of R, Ass1(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 R = 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 uR has no irrelevant prime divisor. In the second main section, we prove that every non-zero finitely generated ideal in a Pr¨ufer domain has the persistence property with respect to weakly associated prime ideals. Finally, we extend the notion of persistence property of ideals to the persistence property for rings.

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 $Qin Spec_L(M)$. 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.

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.

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.

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.