Generalizations of Primary Ideals in Commutative Rings

Mathematics

In this paper, we introduce the concepts of almost right primary ideals and almost nilary ideals and study their related results. We compare almost right primary ideals with other types of ideals, such as right primary ideals and weakly right primary ideals, and investigate their forms in decomposable rings. Moreover, we study the prime radical of an ideal of the product rings. Finally, we provide a definition of fully almost right primary rings and demonstrate that the homomorphic image of a fully almost right primary ring is again a fully almost right primary ring. We also investigate the quotient structure of fully almost right primary rings.

TURKISH JOURNAL OF MATHEMATICS, 2016

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.

Quaestiones Mathematicae, 2020

Let A be an integral domain with quotient field K. A. Badawi and E. Houston called a strongly primary ideal I of A if whenever x, y ∈ K and xy ∈ I, we have x ∈ I or y n ∈ I for some n ≥ 1. In this note, we study the generalization of strongly primary ideal to the context of arbitrary commutative rings. We define a primary ideal P of A to be strongly primary if for each a, b ∈ A, we have aP ⊆ bA or b n A ⊆ a n P for some n ≥ 1.

arXiv (Cornell University), 2022

We define a new generalization of −absorbing ideals in commutative rings called −absorbing −primary ideals. We investigate some characterizations and properties of such new generalization. If is an −absorbing −primary ideal of and √ = √ , then √ is a −absorbing −primary ideal of. And if √ is an (− 1) −absorbing ideal of such that � √ ⊆ , then is an −absorbing −primary ideal of .

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...

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.

Let R be a commutative ring with identity and all modules are unital. Various generalizations of primary ideals and primary modules have been studied. For example, a proper ideal I of R is weakly primary if whenever 0=ab∈I, then a ∈I or b∈ Rad (I). Also a proper submodule N of R-module M is weakly primary submodule if whenever r∈R and m∈M such that 0≠rm∈N , then either m ∈N or r n ∈(N:M). Through out this work, we define almost primary ideals and almost primary submodules as a new generalization of weakly primary ideals (resp., primary submodules). We show that weakly primary ideals (resp., primary submodules) enjoy analogs of many of the properties of primary ideals (resp., submodules).

All rings are commutative with $1$ and $n$ is a positive integer. Let $\phi: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $\phi$-$n$-absorbing primary if whenever $a_1,a_2,...,a_{n+1}\in R$ and $a_1a_2\cdots a_{n+1}\in I\backslash\phi(I)$, either $a_1a_2\cdots a_n\in I$ or the product of $a_{n+1}$ with $(n-1)$ of $a_1,...,a_n$ is in $\sqrt{I}$. The aim of this paper is to investigate the concept of $\phi$-$n$-absorbing primary ideals.

Communications in Algebra, 2008

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.