Let R be a domain with quotiont field K, and let N be a SUBMODULE of an R -module M. We say that N is powerful (strongly primary) if x, yÎK and xyMÍN, then xÎR or yÎR (xMÍN or ynMÍN for some n³1). We show that a SUBMODULE with either of these properties is comparable to every PRIME SUBMODULE of M, also we show that an R -module M admits a powerful SUBMODULE if and only if it admits a strongly primary SUBMODULE. Finally we study finitely generated torsion free modules over domain each of whose PRIME SUBMODULEs are strongly primary.

Let $R= \bigoplus_{g \in G} R_g$ be a $G-$graded commutative ring with identity, $I$ be a graded ideal and let $M$ a $G-$graded unitary $R$-module, where $G$ is a semigroup with identity $e$. We introduce graded $I-$PRIME ideals (SUBMODULEs) as a generalizations of the classical notions of PRIME ideals (SUBMODULEs). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module $F$, a graded SUBMODULE $P$ of $M$ is $I-$PRIME if and only if $F \otimes P$ is graded $I-$PRIME SUBMODULE of $F \otimes M$. As an application, for finitely generated graded module $M$ over Noetherian graded ring $R$, the completion of graded $I-$PRIME SUBMODULEs is $I-$PRIME SUBMODULE.

Let $L$ be a complete lattice. Let $R$ be a commutative ring, $M$ an $R$-module and $\nu$ an $L$-SUBMODULE of $M$. $\nu$ is called a classical PRIME $L$-SUBMODULE of $M$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that either $a_rx_t\in \nu$ or $b_sx_t\in \nu$. Assume that $\nu$ is an $L$-SUBMODULE of $mmu\in L(M)$. We say that $\nu$ is a $2$-absorbing $L$-SUBMODULE of $\mu$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that $a_rb_s\mu\subseteq \nu$ or $a_rx_t\in \nu$ or $b_sx_t\in \nu$. In this case every PRIME $L$-SUBMODULE of $M$ is a classical PRIME $L$-SUBMODULE, and every classical PRIME $L$-SUBMODULE is a $2$-absorbing $L$-SUBMODULE. In this paper we give some basic results concerning these classes of $L$-SUBMODULEs. Finally we topologize $L-Cl. Spec(M)$, the set of all classical PRIME $L$-SUBMODULEs of $M$, with Zariski topology.

Let R be a commutative ring with identity and M be a unital R-module. A proper SUBMODULE N of M with N: R M = p is said to be PRIME or p-PRIME (p a PRIME ideal of R) if rx 2 N for r 2 R and x 2 M implies that either x 2 N or r 2 p. In this paper we study a new equivalent conditions for a minimal PRIME SUBMODULEs of an R-module to be a  nite set, whenever R is a Noetherian ring. Also we introduce the concept of arithmetic rank of a SUBMODULE of a Noetherian module and we give an upper bound for it.

Let G be a group with identity e. Let R be a G-graded commutative ring with identity 1 and M a graded R-module. A proper graded SUBMODULE C of M is called a graded classical PRIME SUBMODULE if whenever r,s 2 h(R) and m 2 h(M) with rsm 2 C, then either rm 2 C or sm 2 C. In this paper, we introduce the concept of graded Jgr-classical PRIME SUBMODULE as a new generalization of graded classical SUBMODULE and we give some results concerning such graded modules. We say that a proper graded SUBMODULE N of M is a graded Jgr-classical PRIME SUBMODULE of M if whenever rsm 2 N where r,s 2 h(R) and m 2 h(M), then either rm 2 N + Jgr(M) or sm 2 N + Jgr(M), where Jgr(M) is the graded Jacobson radical.

The generalized principal Ideal theorem is one of the cornerstones of the dimension theory for the Noetherian rings. For an R-module M, certain SUBMODULEs of M that play a role analo- gous to that of PRIME ideals in the ring R are identied. Using this denition, we extend the this theorem to modules.

Today, literature and poetry have a more comprehensive definition than mere literary expression with academic meanings and tools. Literature is the collection of all the scientific abilities of the language in the field of structure and richness. The artistic creativity of the poet and the employment of excellent compositions and themes lead to the production of a literary and artistic work, which can be found in the excitement and expansion of the audience. The aim of this research is to reach the communication network of all the factors related to the language and the appearance of the word, which not only carries the meaning, but also creates a different meaning and concept from the apparent meaning that results from the phonetic, lexical or visual mechanism. A network that is referred to as "artwork". Artistic device (PRIME) is a meta-array approach to the aesthetic tools of language in poetry, which formalists consider to be the art of alienating language from the mind of the audience.

An $R$-module $M$ is called torsion-free, if $rx=0$ for $r\in R$ and $x\in M$ implies that $r=0$ or $x=0$. In this paper, we introduce the notions semi torsion-free modules and quasi torsion-free modules. We show that a SUBMODULE $N$ of an $R$-module $M$ is a $P$-primary SUBMODULE if and only if $\dfrac{R}{P}$-module $\dfrac{M}{N}$ is semi torsion-free. Also we define a new radical in free modules and find some characterizations of it. We prove that for $P$-SUBMODULE $N$ of a free $R$-module $F$ which $\sqrt N \subsetneqq F$, we have for any $r \in R$ and $m \in F$, $rm \in N$ implies $r \in \sqrt P$ or $m \in \sqrt N$ if and only if $\dfrac{R}{P}$-module $\dfrac{F}{N}$ is quasi torsion free.