Abstract:
A module over an associative ring with unity is a QTAG module if every finitely generated submodule of any homomorphic image of is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which -pure submodules and high submodules are significant. A submodule is quasi- -dense in if is -divisible, for every -pure submodule of containing Here we study these submodules and obtain some interesting results. Motivated by -neat envelope, we also define -pure envelope of a submodule as the -pure submodule if has no direct summand containing We find that -pure envelopes of have isomorphic basic submodules, and if is the direct sum of uniserial modules, then all -pure envelopes of are isomorphic. 1. Introduction All the rings considered here are associative with unity, and right modules are unital modules. An element is uniform, if is a nonzero uniform (hence uniserial) module and for any -module with a unique decomposition series, denotes its decomposition length. For a uniform element , , and are the exponent and height of in , respectively. denotes the submodule of generated by the elements of height at least , and is the submodule of generated by the elements of exponent at most . ？ is -divisible if , and it is -reduced if it does not contain any -divisible submodule. In other words, it is free from the elements of infinite height. The modules , form a neighbourhood system of zero giving rise to -topology. The closure of a submodule is defined as , and it is closed with respect to -topology if . A submodule of is -pure in if , for every integer . For a limit ordinal , , for all ordinals , and it is -pure in if for all ordinals . A module is summable if , where is the set of all elements of which are not in , where is the length of . A submodule is nice [1, Definition 2.3] in , if for all ordinals ; that is, every coset of modulo may be represented by an element of the same height. The cardinality of the minimal generating set of is denoted by . For all ordinals , is the - invariant of and it is equal to . For a module , there is a chain of submodules , for some ordinal . , where is the submodule of . Singh [2] proved that the results which hold for TAG modules also hold good for modules. 2. Quasi- -Dense Submodules In [3], we studied semi- -pure submodules which are not -pure but contained in -pure submodules. Now we investigate the submodules such that is -divisible for every -pure submodule , containing . These modules are called quasi- -dense submodules. We start with the following. Definition 1. A submodule of is quasi-

Abstract:
The concept of a QTAG-module MR was given by Singh [8]. The structure theory of such modules has been developed on similar lines as that of torsion abelian groups. If a module MR is such that M ￠ M is a QTAG-module, it is called a strongly TAG-module. This in turn leads to the concept of a primary TAG-module and its periodicity. In the present paper some decomposition theorems for those primary TAG-modules in which all h-neat submodules are h-pure are proved. Unlike torsion abelian groups, there exist primary TAG-modules of infinite periodicities. Such modules are studied in the last section. The results proved in this paper indicate that the structure theory of primary TAG-modules of infinite periodicity is not very similar to that oftorsion abelian groups.

Abstract:
Let R be a commutative ring with nonzero identity. Our objective is to investigate representable modules and to examine in particular when submodules of such modules are representable. Moreover, we establish a connection between the secondary modules and the pure-injective, the Σ-pure-injective, and the prime modules.

Abstract:
Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. Let $\mathcal{S}(M)$ be the set of all submodules of $M$, and $\phi:\mathcal{S}(M)\to \mathcal{S}(M)\cup \{\emptyset\}$ be a function. We say that a proper submodule $P$ of $M$ is a prime submodule relative to $\phi$ or $\phi$-prime submodule if $a\in R$, $x\in M$ with $ax\in P\setminus \phi(P)$ implies that $a\in(P:_RM)$ or $x\in P$. So if we take $\phi(N)=\emptyset$ for each $N\in\mathcal{S}(M)$, then a $\phi$-prime submodule is exactly a prime submodule. Also if we consider $\phi(N)=\{0\}$ for each submodule $N$ of $M$, then in this case a $\phi$-prime submodule will be called a weak prime submodule. Some of the properties of this concept will be investigated. Some characterizations of $\phi$-prime submodules will be given, and we show that under some assumptions prime submodules and $\phi_1$-prime submodules coincide.

Abstract:
Let R be a ring and M a right R-module. It is shown that (1) δ(M) is Noetherian if and only if M satisfies ACC on δ-small submodules; (2) δ(M) is Artinian if and only if M satisfies DCC on δ-small submodules; (3) M is Artinian if and only if M is an amply δ-supplemented module and satisfies DCC on δ-supplement submodules and on δ-small submodules.

Abstract:
We introduce the concept of almost semiprime submodules of unitary modules over a commutative ring with nonzero identity. We investigate some basic properties of almost semiprime and weakly semiprime submodules and give some characterizations of them, especially for (finitely generated faithful) multiplication modules. 1. Introduction Throughout this paper, all rings are commutative rings with identity and all modules are unitary. Various generalizations of prime (primary) ideals are studied in [1–8]. The class of prime submodules of modules as a generalization of the class of prime ideals has been studied by many authors; see, for example, [9, 10]. Then many generalizations of prime submodules were studied such as weakly prime (primary) [11], almost prime (primary) [12], 2-absorbing [13], classical prime (primary) [14, 15], and semiprime submodules [16]. In this paper, we study weakly semiprime and almost semiprime submodules as the generalizations of semiprime submodules. Weakly semiprime submodules have been already studied in [17]. Here we first define the notion almost semiprime submodules and get a number of propensities of almost semiprime and weakly semiprime submodules. Also, we give some characterizations of such submodules in multiplication modules. Now we define the concepts that we will use. For any two submodules and of an -module , the residual of by is defined as the set which is clearly an ideal of . In particular, the ideal is called the annihilator of . Let be a submodule of and let be an ideal of ; the residual submodule of by is defined as . These two residual ideals and submodules were proved to be useful in studying many concepts of modules; see, for example, [18, 19]. A proper submodule of an -module is a prime submodule if, whenever for and , or . An -module is called a prime module if its zero submodule is a prime submodule. A proper submodule of an -module is called weakly prime (weakly primary) if , where and ; then or ( or ). A proper submodule of an -module is called almost prime (almost primary) if, whenever for and , or ( or ). A proper ideal of a commutative ring is called semiprime if , where and ; then . A proper submodule of an -module is called semiprime if, whenever , , and such that , . An -module is called a second module provided that, for every element , the -endomorphism of produced by multiplication by is either surjective or zero; this implies that is a prime ideal of and is said to be -second [20]. An -module is called a multiplication module provided that, for every submodule of , there exists an ideal of

Abstract:
Let $R$ be a commutative ring and let $M$ be an $R$-module. A submodule $N$ of $M$ is called a weakly primal submodule provided that the set $ P = w(N) cup { 0 } $ forms an ideal of $R$. Here $w(N)$ is the set of elements of $R$ that are not weakly prime to $N$, where an element $ r in R $ is not weakly prime to $N$ if $ 0 eq rm in N $ for some $ m in M ackslash N $. In this paper we give some basic results about weakly primal submodules. Also we discuss on the relations between the classes of the weakly primal submodules of $M$ and the weakly primal submodules of modules of fractions of $M$.

Abstract:
By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results.

Abstract:
Let $ R $ be a commutative ring with non-zero identity. We define a proper submodule $ N $ of an $ R $-module $ M $ to be weakly prime if $ 0 ot = rmin N $( $ rin R, min M $) implies $ min N $ or $ rMsubseteq N $. A number of results concerning weakly prime submodules are given. For example, we give three other characterizations of weakly prime submodules.

Abstract:
We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if $M$ is a finitely generated quasi-projective left $R$-module with $S=End_R(M)$ and $N$ is an $M$-generated left $R$-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left $R$-submodules of $N$ and coclosed left $S$-submodules of $Hom_R(M,N)$.