On Weakly 1-absorbing Primary Ideals of Commutative Rings

2020

Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ 0. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN -rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.

Journal of the Korean Mathematical Society, 2015

Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, c ∈ R and 0 = abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

2015

Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, c ∈ R and 0 = abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

Journal of Algebra and Its Applications

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] fo...

Filomat, 2022

An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a,b ? R and 0 ? ab ? I, then a ? I or b ? ?I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ? R and 0 ? abc ? I, then ab ? I or c ? ?I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary.

All rings are commutative with 1 = 0. A proper ideal I of a ring R is said to be 2-absorbing if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. A proper ideal I of R is said to be 2-absorbing primary if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ I or ac ∈ √ I or bc ∈ √ I. Moreover, a proper ideal I of R is a weakly 2-absorbing primary ideal if whenever a, b, c ∈ R and 0 = abc ∈ I, then either ab ∈ I or ac ∈ √ I or bc ∈ √ I. In this study we give some characterizations of 2-absorbing primary and weakly 2-absorbing primary ideals.

Bulletin of the Korean Mathematical Society, 2014

Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of 2-absorbing primary ideal which is a generalization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

2016

h t t p : / / j o u r n a l s. t u b i t a k. g o v. t r / m a t h / Abstract: Let R be a commutative ring with 1 ̸ = 0 and S(R) be the set of all ideals of R. In this paper, we extend the concept of 2-absorbing primary ideals to the context of ϕ-2-absorbing primary ideals. Let ϕ : S(R) → S(R) ∪ ∅ be a function. A proper ideal I of R is said to be a ϕ-2-absorbing primary ideal of R if whenever a, b, c ∈ R with abc ∈ I − ϕ(I) implies ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning ϕ-2-absorbing primary ideals are given. weakly 2-absorbing ideal, 2-absorbing primary ideal, weakly 2-absorbing primary ideal, ϕ-prime ideal, ϕ-2-primary ideal, ϕ-2-absorbing ideal

arXiv (Cornell University), 2020

Let R be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ I. Some properties of 1-absorbing primary ideals are investigated. For example, we show that if R admits a 1-absorbing primary ideal that is not a primary ideal, then R is a quasilocal ring. We give an example of a 1-absorbing primary ideal of R that is not a primary ideal of R. We show that if a ring R is not a quasilocal, then a proper ideal I of R is a 1-absorbing primary ideal of R if and only if I is a primary ideal. We show that if R is a Noetherian domain, then R is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of R is of the form P n for some nonzero prime ideal P of R and a positive integer n ≥ 1. We show that a proper ideal I of R is a 1-absorbing primary ideal of R if and only if whenever I 1 I 2 I 3 ⊆ I for some proper ideals I 1 , I 2 , I 3 of R, then I 1 I 2 ⊆ I or I 3 ⊆ √ I.

2013

Let R be a commutative ring with identity 1 = 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a, b ∈ R with 0 = ab ∈ I, then either a ∈ I or b ∈ I. Also a proper ideal I of R is said to be 2-absorbing if whenever a, b, c ∈ R and abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a, b, c ∈ R and 0 = abc ∈ I, then either ab ∈ I or ac ∈ I or bc ∈ I. For example, every proper ideal of a quasi-local ring (R, M ) with M 3 = {0} is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I 3 = 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 = {0} or R is ringisomorphic to R 1 × F where R 1 is a quasi-local ring with maximal ideal M such that M 2 = {0} and F is a field or R is ring-isomorphic to F 1 × F 2 × F 3 for some fields F 1 , F 2 , F 3 .