In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

Let R be a commutative ring and M an R -module. In this article, we introduce the concept of S -2-absorbing submodule. Suppose that S ⊆ R is a multiplicatively closed subset of R. A submodule P of M with ( P : _R M ) ∩ S = ∅ is said to be an S -2-absorbing submodule if there exists an element s ∈ S and whenever abm ∈ P for some a, b ∈ R and m ∈ M , then sab ∈ ( P : _R M ) or sam ∈ P or sbm ∈ P. Many examples, characterizations and properties of S -2-absorbing submodules are given. Moreover, we use them to characterize von Neumann regular modules in the sense [9].

Let R be a commutative ring with nonzero identity, and let M be a nonzero unital R-module. In this article, we introduce the concept of 2-absorbing quasi primary submodules which is a generalization of prime submodules. We define 2-absorbing quasi primary submodule as a proper submodule N of M having the property that abm ? N, then ab ? ?(N :R M) or am ? radM(N) or bm ? radM(N): Various results and examples concerning 2-absorbing quasi primary submodules are given.

Recall that a ring R is said to be a quasi regular ring if its total quotient ring q(R) is von Neumann regular. It is well known that a ring R is quasi regular iff it is a reduced ring satisfying the property: for each a \in R, ann(ann(a)) = ann(b) for some b \in R. Here, in this study, we extend the notion of quasi regular rings and rings which satisfy the aforementioned property to modules. We give many characterizations and properties of these two classes of modules. Moreover, we investigate the (weak) quasi regular property of trivial extension.