On graded coherent-like properties in trivial ring extensions

On graded 1 -absorbing δ -primary ideals

“…For an ideal I of A and a submodule F of E, I ∝ F is a graded ideal of A ∝ E if and only if I is a graded ideal of A and F is a graded submodule of E and IE ⊆ F , see [10,Proposition 3.3]. This result is furtherly generalized to the case where G is a commutative monoid (not necessarily an abelian group), see [4,Theorem 2]. The authors in [10] determined the certain classes of graded ideals such as graded maximal ideals, graded prime ideals, graded primary ideals, graded quasi primary ideals, graded 2-absorbing ideals and graded 2-absorbing quasi primary ideals of graded idealization A ∝ E. Now, we investigate the graded 1-absorbing δ-primary ideals in A ∝ E.…”

On graded pseudo 2-prime ideals

“…An R-submodule N of M is said to be a graded submodule if whenever m = P g∈G m g ∈ N then m g ∈ N for all g ∈ G. Let R be a G-graded ring and M be a G-graded R-module. The graded trivial extension of R-module M is denoted by RM = R ⊕ M with componentwise addition and the following multiplication (a, m)(b, m 0 ) = (ab, am 0 + bm) for all a, b ∈ R and m, m 0 ∈ M. Then RM is a G-graded ring by (RM ) g = R g ⊕ M g for all g ∈ G [19]. Suppose that I is an ideal of R and N is a submodule of M .…”

“…Suppose that I is an ideal of R and N is a submodule of M . Then IN(= I ⊕ N additively) is a graded ideal of RM if and only if I is a graded ideal of R, N is a graded submodule of M and IM ⊆ N [19,Theorem 2.4]. For more informations and other terminology on graded rings and modules, we refer [20] to the reader.…”