Quasi J-ideals of commutative rings

(m, n)-Closed Submodules of Modules over Commutative Rings

“…For example, the ideal I = 16Z is (3,2)-closed in the ring of integers which is not (3, 2)-prime as 2 3 • 2 ∈ I but 2 2 , 2 / ∈ I. For more classes of ideals related to (m, n)-closed ideals, one can see [14] and [15].…”

“…Since r m j b ∈ N j , then by assumption r n j ∈ (N j : R M) and so r ∈ (N j : R M) = P as n ≥ n j . By [13, Lemma 1], we have for all i ∈ {1, In general, if N 1 and N 2 are two (m, n)-closed submodules of an R-module M with (N 1 : R M) ̸ = (N 2 : R M), then N 1 ∩ N 2 need not be (m, n)-closed, see [13], Remark 2.…”

“…According to [3], a proper ideal I of R is called an (m, n)-closed ideal if whenever r m ∈ I for some r ∈ R, then r n ∈ I. In a recent work [13], the class of (m, n)-prime ideals which is a subclass of the previous structure is defined and studied. A proper ideal I of R is called (m, n)-prime Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s).…”