On graded $A$-2-absorbing submodules of graded modules over graded commutative rings

On graded G2-absorbing and graded strongly G2-absorbing second submodules

“…Let R be a G-graded ring, M a graded R-module, N = ⊕ g∈G N g a graded submodule of M and g ∈ G. Then N g is said to be a strongly g-2-absorbing second submodule of the R e -module M g , if N g 6 = {0} g ; and whenever r e , s e ∈ R e and K g is a submodule of M g with r e s e N g ⊆ K g , then either r e N g ⊆ K g or s e N g ⊆ K g or r e s e ∈ Ann R e (N g ), (see [1]).…”

“…Also, graded secondary modules have been introduced by Atani and Farzalipour in [12]. The concepts of graded 2-absorbing second submodules and graded strongly 2-absorbing second submodules were introduced and studied in [1,18].…”

“…It is shown in [10,Lemma 2.1] that if N is a graded submodule of M , then (N : R M ) is a graded ideal of R. The annihilator of M is defined as (0 : R M ) and is denoted by Ann R (M ). A proper graded submodule C of M is said to be a completely graded irreducible if C = ∩ α∈∆ C α , where {C α } α∈∆ is a family of graded submodules of M , implies that C = C β for some β ∈ ∆, (see [1]). A non-zero graded submodule N of M is said to be a graded second if for each r ∈ h(R), the endomorphism of N given by multiplication by r is either surjective or zero, (see [7]).…”