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 $L$ 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.

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.