scholarly journals On Graded S-Primary Ideals

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2637
Author(s):  
Azzh Saad Alshehry
Keyword(s):  
Primary Ideal ◽  
Graded Ring ◽  
Basic Properties ◽  
Primary Ideals

Let R be a commutative graded ring with unity, S be a multiplicative subset of homogeneous elements of R and P be a graded ideal of R such that P⋂S=∅. In this article, we introduce the concept of graded S-primary ideals which is a generalization of graded primary ideals. We say that P is a graded S-primary ideal of R if there exists s∈S such that for all x,y∈h(R), if xy∈P, then sx∈P or sy∈Grad(P) (the graded radical of P). We investigate some basic properties of graded S-primary ideals.

scholarly journals Hilbert-Samuel function and Grothendieck group

2000 ◽  
Vol 43 (1) ◽  
pp. 73-94
Author(s):  
Koji Nishida
Keyword(s):  
Local Ring ◽  
Grothendieck Group ◽  
Residue Field ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module ◽  
Primary Ideals

AbstractLet (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

scholarly journals The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

2019 ◽  
Vol 74 (1) ◽  
pp. 7-16
Author(s):  
Khaldoun Al-Zoubi ◽  
Malik Jaradat
Keyword(s):  
Commutative Rings ◽  
Primary Ideal ◽  
Zariski Topology ◽  
Graded Ring ◽  
Primary Spectrum ◽  
Primary Ideals

Abstract Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.

scholarly journals On 1-absorbing δ-primary ideals

2021 ◽  
Vol 29 (3) ◽  
pp. 135-150
Author(s):  
Abdelhaq El Khalfi ◽  
Najib Mahdou ◽  
Ünsal Tekir ◽  
Suat Koç
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Basic Properties ◽  
New Class ◽  
Ring Extensions ◽  
Primary Ideals ◽  
Nonzero Identity

Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

scholarly journals Characterizations of Graded Rings Over Which Every Graded Semi-primary Ideal is Graded 1-absorbing Primary

2021 ◽  
Author(s):  
Alaa Melhem ◽  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Primary Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Primary Ideals

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. Graded semi-primary and graded $1$-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we characterize graded rings over which every graded semi-primary ideal is graded $1$-absorbing primary and graded rings over which every graded $1$-absorbing primary ideal is graded primary.

scholarly journals Characterizations of Graded Rings over Which Every Graded semi-primary Ideal is Graded 1-Absorbing Primary

2021 ◽  
Vol 20 ◽  
pp. 547-553
Author(s):  
Alaa Melhem ◽  
Malik Bataineh ◽  
Rashid Abu-Dawwas
Keyword(s):  
Primary Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Primary Ideals

Let G be a group with identity e and R be a commutative G-graded ring with nonzero unity 1. Graded semi-primary and graded 1-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we character­ ize graded rings over which every graded semi-primary ideal is graded 1-absorbing primary and graded rings over which every graded 1-absorbing primary ideal is graded primary.

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
Author(s):  
Ayman Badawi ◽  
Ece Yetkin Celikel
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Primary Ideals ◽  
Nonzero Identity

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] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

scholarly journals Notes on irreducible ideals

1985 ◽  
Vol 31 (3) ◽  
pp. 321-324
Author(s):  
David J. Smith
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Primary Ideal ◽  
Regular Local Ring ◽  
Finite Intersection ◽  
Irreducible Components ◽  
Primary Ideals

Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

scholarly journals On 2-Absorbing Primary Submodules of Modules over Commutative Rings

2016 ◽  
Vol 24 (1) ◽  
pp. 335-351 ◽  
Author(s):  
Hojjat Mostafanasab ◽  
Ece Yetkin ◽  
Ünsal Tekir ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Rings ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Prime Submodule ◽  
Primary Ideals

Abstract All rings are commutative with 1 ≠ 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a; b ∈ R and m ∈ M and abm ∈ N, then am ∈ M-rad(N) or bm ∈ M-rad(N) or ab ∈(N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1; I2 of R and some submodule K of M, then I1I2 ⊆ (N :R M) or I1K ⊆ M-rad(N) or I2K ⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N :R M) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.

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