Commutative Ring Research Articles

Bipartite through Prescribed Median and Antimedian of a Commutative Ring with Respect to an Ideal

Bipartite through Prescribed Median and Antimedian of a Commutative Ring with Respect to an Ideal

Asymptotic behavior of homological invariants of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M/InM at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals.

Computing the closure of a support

When E is an R-module over a commutative unital ring R, the Zariski closure of its support is of the form V ( O ( E ) ) where O ( E ) is a unique radical ideal. We give an explicit form of O ( E ) and study its behavior under various operations of algebra. Applications are given, in particular for ring extensions of commutative unital rings whose supports are closed. We provide some applications to crucial and critical ideals of ring extensions.

Modules and rings satisfying strong accr

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an [Formula: see text]-module. We say that [Formula: see text] satisfies strong accr[Formula: see text] if for every submodule [Formula: see text] of [Formula: see text] and for every sequence [Formula: see text] of elements of [Formula: see text] the ascending sequence of submodules of the form, [Formula: see text] is stationary. We say that a ring [Formula: see text] satisfies strong accr[Formula: see text] if [Formula: see text] regarded as a module over [Formula: see text] satisfies strong accr[Formula: see text] In this paper, we give a necessary and sufficient condition for the pulback (respectively, the Nagata’s idealization [Formula: see text]) to be strong accr[Formula: see text] ring. We also prove a new characterization for a valuation ring to be strong accr[Formula: see text]

Classical 1-Absorbing Primary Submodules

Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let ℜ be a commutative ring and M an ℜ-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if xyzη∈K for some η∈M and nonunits x,y,z∈ℜ, then xyη∈K or ztη∈K for some t≥1. In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules.

Two absorbing factorization formal power series rings

Let R be a commutative ring with identity. A proper ideal I of R is said to be 2-absorbing if whenever xyz ∈ I for some elements x , y , z ∈ R , then either xy ∈ I or xz ∈ I or yz ∈ I . The ring R is called a two-absorbing factorization ring (TAF-ring) if every proper ideal of R has a two absorbing-factorization that is every ideal is a product of 2-absorbing ideals. In this note, we characterize commutative rings R (respectively, commutative ring extensions A ⊂ B ) for which the ring of formal power series R [ [ X ] ] (respectively, the ring A + XB [ [ X ] ] ) is a TAF-ring.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

The finite type of modules of bounded projective dimension and Serre's conditions

AbstractLet be a commutative Noetherian ring. For a natural number , we prove that the class of modules of projective dimension bounded by is of finite type if and only if satisfies Serre's condition . In particular, this answers positively a question of Bazzoni and Herbera in the specific setting of a Gorenstein ring. Applying similar techniques, we also show that the ‐dimensional version of the Govorov–Lazard theorem holds if and only if satisfies the ‘almost’ Serre condition .

Dynamical systems for arithmetic schemes

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space Wrat(X) to every scheme X. We also define R-valued points Wrat(X)(R) of Wrat(X) for every commutative ring R. For normal schemes X of finite type over specZ, using Wrat(X)(ℂ) we construct infinite dimensional R-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of Wrat(X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues–Fontaine curve.

On the graph $$G_P(R)$$ over commutative ring R

On the graph $$G_P(R)$$ over commutative ring R

Metric and Upper Dimension of Extended Annihilating-Ideal Graphs

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph [Formula: see text] associated to a commutative ring [Formula: see text], denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Here we prove the finiteness conditions of [Formula: see text] and [Formula: see text]. In addition, we characterize [Formula: see text], [Formula: see text] and [Formula: see text] for artinian rings and the direct product of rings.

Randić spectrum of the weakly zero-divisor graph of the ring ℤn

In this article, we find the Randić spectrum of the weakly zero-divisor graph of a finite commutative ring R with identity 1 ≠ 0 , denoted as W Γ ( R ) , where R is taken as the ring of integers modulo n . The weakly zero-divisor graph of the ring R is a simple undirected graph with vertices representing non-zero zero-divisors in R . Two vertices, denoted as a and b, are connected if there are elements x in the annihilator of a and y in the annihilator of b such that their product xy equals zero. In particular, we examine the Randić spectrum of W Γ ( Z n ) for specific values of n , which are products of prime numbers and their powers.

On S-2-Prime Ideals of Commutative Rings

Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S-prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S-2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S-prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S-2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S-properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S-prime ideals and 2-prime ideals can be readily expanded to the framework of S-2-prime ideals. Supporting examples also highlight a significant distinction between S-2-prime ideals and stated ideals.

Pythagorean fuzzy nil radical of Pythagorean fuzzy ideal

In this work, we introduce the Pythagorean fuzzy nil radical of a Pythagorean fuzzy ideal of a commutative ring, we further provide the notion of Pythagorean fuzzy semiprime ideal, and we study some related properties. Finally, we give the relation between Pythagorean fuzzy semiprime ideals and the Pythagorean fuzzy nil radical of a commutative ring.

The principal small intersection graph of a commutative ring

The principal small intersection graph of a commutative ring

Matrices of the Zero Divisor Graphs of Classes of 3-Radical Zero Completely Primary Finite Rings

The study of finite completely primary rings through the zero divisor graphs, the unit groups and their associated matrices, and the automorphism groups have attracted much attention in the recent past. For the Galois ring R′ and the 2-radical zero finite rings, the mentioned algebraic structures are well understood. Studies on the 3-radical zero finite rings have also been done for the unit groups and the zero divisor graphs Γ(R). However, the characterization of the matrices associated with these graphs has not been exhausted. It is well known that proper understanding of the classification of zero divisor graphs with diameter 2 and girth 3 can provide insights into the structure of commutative rings and their zero divisors. In this study, we consider a class of 3-radical zero completely primary finite rings whose diameter and girth are 2 and 3 respectively. We enhance the understanding of the structure of such rings by investigating their Adjacency, Laplacian and Distance matrices.

On functional prime ideals in commutative rings

In this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ ( m ) ∈ I , then a ∈ I or ϕ ( m ) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R ( I , R ) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R ( I − 1 , R ) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided.

On graded pseudo 2-prime ideals

In this paper, we study graded pseudo 2-prime ideals of graded commutative rings with nonzero identities. Let G be a commutative additive monoid with an identity element 0, and R=⊕g∈G Rg be a commutative graded ring with a nonzero identity element. A proper graded ideal I of R is said to be a graded pseudo 2-prime ideal if whenever ab ∈ I for some homogeneous elements a,b ∈ R, then a²ⁿ ∈ Iⁿ or b²ⁿ ∈ Iⁿ for some n ∈ ℕ. Besides giving many properties of graded pseudo 2-prime ideals, we characterize graded almost valuation domains in terms of our new concept.

Existence and uniqueness of S-primary decomposition in S-Noetherian modules

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with ( N : R M ) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules.

When every S-flat module is (flat) projective

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS , is a flat RS -module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the “S-perfectness” for various ring constructions, which allows the construction of more interesting examples.