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.

The aim of this paper is to study unitary regular modules on commutative rings with identity. Regularity accompanied by cocyclic property results in some Prime-related conclusions on both modules and rings. Further to this, regularity addresses also radical property of submodules and they are related closely. This property not only affects the modules on ring R but also restricts R to totally idempotent one.