Let R be a commutative ring with identity. A proper submod-ule N of an R-module M is an n-submodule if rm 2 N (r 2 R; m 2 M) with r =2 p AnnR(M), then m 2 N. A number of results concerning n-submodules are given. For example, we give other characterizations of n-submodules. Also various properties of n-submodules are considered.

Let MR be a module with S=End (MR). We call a submodule K of MR annihilator-small if K+T=M, T a submodule of MR, implies that ls (T) =0, where lS indicates the left annihilator of T over S. The sum AR (M) of all such submodules of MR contains the Jacobson radical Rad (M) and the left singular submodule ZS (M). If MR is cyclic, then AR (M) is the unique largest annihilator-small submodule of MR. We study AR (M) and KS (M) in this paper. Conditions when AR (M) is annihilator-small and KS (M) =J (S) =Tot (M; M) are given.

Let R be a commutative ring with non-zero identity and M be a unitary R-module. Then the concept of quasi-secondary submodules of M is introduced and some results concerning this class of submodules is obtained.

LetG be a group. Let R be a G -graded commutative ring with identity, and let M be a G -graded module over R. Two graded submodulesN and K of graded module M are called graded coprime whenever N+K=M. In this paper, some properties of graded coprime submodules are discussed. For example, we show that ifM is a graded finitely generated module, then two graded submodulesN and K of M are graded coprime if and only if gradM (N) and gradM (K)are graded coprime.

Let M and P be right R-modules. A submodule K of an R-module M is called P-dense if for each m Î M; (K: m) is a P-faithful right ideal of R: PR is nonsingular if and only if, for each R-module M; every essential submodule of M is a P-dense submodule. For any R-module M, we obtain P-rational extention of M and equivalent condition in order that M is equal with its P-rational extention is found. An R-module P is called right Kasch if every simple R-module can be embedded in P: Finally, we given some equivalent conditions for an R-module P to be right Kasch.

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.

The submodules with the property of the title (a submodule N of an R -module M is called strongly dense in M, denoted by N£sd M, if for any index set I, ΠIN£d ΠIM) are introduced and fully investigated. It is shown that for each submoduleN of M there exists the smallest subsetD′ ÍM such that N+D′ is a strongly dense submodule of M and D′ ∩ N=0. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module M, dense submodules in M are strongly dense if and only if M £sd ~E (M), where ~E (M) is the rational hull of M. It is proved that R has no strongly dense left ideal if and only if no nonzeroelement of every cyclic R -module M has a strongly dense annihilator in R. Finally, some properties and new concepts related to strong density are studied.