J-Ideals of Commutative Rings

Abstract

Let R be a commutative ring with identity and N (R) and J (R) denote the nilradical and the Jacobson radical of R, respectively. A proper ideal I of R is called an n-ideal if for every a, b ∈ R, whenever ab ∈ I and a / ∈ N (R), then b ∈ I. In this paper, we introduce and study J-ideals as a new generalization of n-ideals in commutative rings. A proper ideal I of R is called a J-ideal if whenever ab ∈ I with a / ∈ J (R), then b ∈ I for every a, b ∈ R.We study many properties and examples of such class of ideals. Moreover, we investigate its relation with some other classes of ideals such as r-ideals, prime, primary and maximal ideals. Finally, we, more generally, define and study J-submodules of an R-modules M . We clarify some of their properties especially in the case of multiplication modules.

Weakly J-ideals of commutative rings

“…Since Ker( f ) ⊆ I 1 , we have 0 xy ∈ I 1 which implies x ∈ J(R 1 ) or y ∈ I 1 . Thus, a = f (x) ∈ J(R 2 ) by [8,Lemma 2.22] or b = f (y) ∈ f (I 1 ) and we are done. Corollary 2.16.…”

“…Lemma 2.19. [9,Theorem 5] Let I be a proper ideal of a ring R. Then I is a quasi J-ideal of R if and only if I ⊆ J(R) and R/I is quasi presimplifiable. Proposition 2.20.…”

“…A proper ideal I of a ring R is called an n-ideal if whenever a, b ∈ R and ab ∈ I such that a N(R), then b ∈ I. Recently, Khashan and Bani-Ata in [8] introduced the notion of J-ideals as a generalization of n-ideals in commutative rings, as follows: A proper ideal I of a ring R is called a J-ideal if whenever a, b ∈ R with ab ∈ I and a J(R), then b ∈ I. In [9], the authors generalized J-ideals and defined quasi J-ideals as those ideals I for which √ I = {x ∈ R : x n ∈ I for some n ∈ N} are J-ideals.…”

“…Recently, Khashan and Bani-Ata in [8] introduced the notion of J-ideals as a generalization of n-ideals in commutative rings, as follows: A proper ideal I of a ring R is called a J-ideal if whenever a, b ∈ R with ab ∈ I and a J(R), then b ∈ I. In [9], the authors generalized J-ideals and defined quasi J-ideals as those ideals I for which √ I = {x ∈ R : x n ∈ I for some n ∈ N} are J-ideals. In this paper, we define and study weakly J-ideals of commutative rings as a new generalization of J-ideals.…”

“…Recall from [4] (resp. [9]) that a ring R is called presimplifiable (resp. quasi presimplifiable) if whenever a, b ∈ R with a = ab, then a = 0 or b ∈ U(R) (resp.…”