Let R be a commutative ring and let I be an ideal of R. In this paper, we will introduce the notions of 2-absorbing I-prime and 2-absorbing I-second submodules of an R-module M as a generalization of 2-absorbing and strongly 2-absorbing second submodules of M and explore some basic properties of these classes of modules.

Let R be a commutative ring with identity and M be a unitaryR-module. A proper submodule N of M is 2- absorbing if r1, r2, r3 Î R, m Î M with r1r2r3m Î M implies r1r2m Î N or r1r3m Î N or r2r3m Î N. Let j: S (M) ® S (M) È {f} be a function where S (M) is the set of all submodules of M. We call a proper submodule No f M a j-2-absorbing submodule if r1, r2, r3 Î R, m Î M with r1r2r3m Î N -j (N) implies that r1r2m Î N or r1r3m Î N or r2r3m Î N. We want to extend 2-absorbing ideals to j-2-absorbing submodules and we show that j-2-absorbing submodules enjoy analogs of many of the properties of 2-absorbing ideals.

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we introduce the concepts of graded semi Jgr-2-absorbing and graded weakly semi Jgr-2-absorbing submodules of M and study the behavior of these notions under several constructions. A proper graded submodule N of M is said to be a graded semi Jgr-2-absorbing (resp. graded weakly semi Jgr-2-absorbing) submodule of M if whenever rg ∈ h(R) and mh ∈ h(M) with rg2mh ∈ N (resp. 0 ≠ rg2mh ∈ N), then either rgmh ∈ N + Jgr(M) or rg2∈ (N + Jgr(M) :R M), where Jgr(M) is the graded Jacobson radical.

Microwave absorbing materials are usually designed to solve protection against electromagnetic interference in wireless communication systems and high frequency circuit mechanisms. In this research polystyrene (PS) nanocomposites containing various nano-fillers were successfully synthesized. The novelty of this work is comparing of three various nanostructures: non-metallic conductive graphene oxide, magnetic Fe3O4 and semi-conductor zinc oxide were used as additive. The effect of different fillers loading and homogenizer speed on the reflection loss (RL) amount and electromagnetic wave absorption was investigated. In order to investigate particle size and morphology of the nanostructures the scanning electron microscopy (SEM) was used. The frequency range of 5-8 GHz was employed in the investigation of electromagnetic wave absorption properties of nanocomposites using a vector network analyzer and eventually their absorption properties were analyzed and compared. The results indicate that graphene oxide has substantial effect on absorption in compare with the other nanocomposite samples. Increase of homogenizer speed led to a dispersion improvement of nanostructures and absorption. Therefore, the broadening of the microwave absorption band-width is attributed to the suitable dispersion of nanostructures in polymeric matrix.

Let R be a commutative ring with identity, S be a mul-tiplicatively closed subset of R, and let M be an R-module. In this paper, we introduce and investigate some properties of the notion of S-2-absorbing second submodules of M as a generalization of S-second submodules and strongly 2-absorbing second submodules of M. Also, we obtain some results concerning S-2-absorbing submodules of M.

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