On 1-absorbing δ-primary ideals

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.

N Sequence Prime Ideals

In this paper, the concepts of -sequence prime ideal and -sequence quasi prime ideal are introduced. Some properties of such ideals are investigated. The relations between -sequence prime ideal and each of primary ideal, -prime ideal, quasi prime ideal, strongly irreducible ideal, and closed ideal, are studied. Also, the ideals of a principal ideal domain are classified into quasi prime ideals and -sequence quasi prime ideals.

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

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.

On Graded S-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.

On Weakly S-Primary Ideals of Commutative Rings

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

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

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.

Note on strongly quasi-primary ideals

Let [Formula: see text] be a commutative ring with [Formula: see text]. A proper ideal [Formula: see text] of [Formula: see text] is said to be a strongly quasi-primary ideal if, whenever [Formula: see text] with [Formula: see text], then either [Formula: see text] or [Formula: see text]. In this paper, we characterize Noetherian and reduced rings over which every (respectively, nonzero) proper ideal is strongly quasi-primary. We also characterize ring over which every strongly quasi primary ideal of [Formula: see text] is prime. Many examples are given to illustrate the obtained results.

f-Primary Ideals in Semigroups

Right now, the terms left f-Primary Ideal, right f-Primary Idealand f- primary ideals are presented. It is Shown that An ideal U in a semigroup S fulfills the condition that If G, H are two ideals of S with the end goal that f (G) f (H)⊆U and f(H)⊈U then f(G)⊆rf (U)iff f (q), f (r)⊆S , <f (q)><f (r)>⊆U and f (r)⊈U then f (q)⊆rf (U) in like manner it is exhibited that An ideal U out of a semigroup S fulfills condition If G, H are two ideals of S such that f (G) f (H)⊆U and f (G)⊈U then f (H) ⊆rf (U) iff f (q), f (r)⊆S,<f (q)><f (r)>⊆U and f (q)⊈U⇒f (r)⊆rf (U). By utilizing the meanings of left - f- primary and right f- primary ideals a couple of conditions are illustrated It is shown that J is a restrictive maximal ideal in Son the off chance thatrf (U) = J for some ideal U in S at that point J will be a f- primary ideal and Jn is f-primary ideal for some n it is explained that if S is quasi-commutative then an ideal U of S is left f - primary iff right f -primary.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

A generalization of coefficient ideals

In this paper we give a generalization of the coefficient ideals of an m \mathfrak {m} -primary ideal I I in a quasi-unmixed local ring R R with infinite residue field.