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Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings. [ABSTRACT FROM AUTHOR]

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results. [ABSTRACT FROM AUTHOR]

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s ∈ S such that whenever x , y ∈ R and xy ∈ I , then either s x 2 ∈ I or sy ∈ I . Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated. [ABSTRACT FROM AUTHOR]

Let G be an abelian group written addditively and R be a commutative graded ring of type G with identity and m, n be positive integers. The main purpose of this paper is to introduce the class of graded-(m, n)-prime ideals which lies properly between the classes of graded-prime and graded-(m, n)-closed ideals introduced recently by the authors in Ahmed et al. (Moroccan J Algebra Geom Appl 1(2):1-10, 2022). A proper graded ideal I of R is called graded-(m, n)-prime if for some homogeneous elements a , b ∈ R , a m b ∈ I implies either a n ∈ I or b ∈ I . Several characterizations of this new class of graded ideals with several original examples are given. Moreover, we defend the actions of graded-(m, n)-prime ideals in several extensions of graded rings, especially in idealization of graded modules and amalgamation of graded rings and similarly to graded-primary decomposition, we introduce the graded-(m, n)-decomposition of graded ideals and we prove that every graded ideal in a graded-n-Noetherian ring has a graded-(m, n)-decomposition. Finally, the graded-(m, n)-prime avoidance theorem is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and explore a category of rings termed SFP-rings, defined by the property that each S-finite ideal of them is projective. We demonstrate that all hereditary rings inherently qualify as SFP-rings, while conversely, every SFP-ring is inherently semi-hereditary. Our research delves into the application and relevance of this concept across various facets of commutative ring extensions. These include localizations, direct products, trivial ring extensions, pullbacks, and the specific case of amalgamated duplication of a ring along an ideal. A significant outcome of our study is the identification and characterization of unique families of rings: those that are non-hereditary yet qualify as SFP-rings, and those that are semi-hereditary but do not meet the criteria for being SFP-rings. This work not only extends the theoretical understanding of ring structures but also contributes to the broader field of algebraic theory with practical examples and applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring with 1 ≠ 0 . An ideal I of A is said to be a non-J-ideal if it is not contained in the Jacobson radical of A. The ring A is called non-J-Noetherian if each non-J-ideal is finitely generated. We show that many of the properties of Noetherain rings are true for non-J-Noetherian rings. Among other results we show that A is a non-J-Noetherian ring if and only if the power series ring A [ [ X 1 , ⋯ , X n ] ] is non-J-Noetherian. Also, we study the transfert of the non-J-Noetherian property via Nagata's idealization. We investigate the relation between nonnil-Noetherian ring and non-J-Noetherian rings and construct non-trivial (not quasi-local) examples of n-dimensional non-J-Noetherian rings for each n ≥ 2 . [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of rings in which every S-finitely presented ideal is finitely presented called SWC-rings. Hence, a ring is coherent if and only if it is S-coherent and an SWC-ring. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and pullbacks. Our results generate families of examples of non-S-coherent SWC-rings and non-SWC S-coherent rings. [ABSTRACT FROM AUTHOR]

The class of good semigroups is a class of subsemigroups of N h , that includes the value semigroups of rings associated to curve singularities and their blowups, and allows to study combinatorically the properties of these rings. In this paper we give a characterization of almost symmetric good subsemigroups of N h , extending known results in numerical semigroup theory and in one-dimensional ring theory, and we apply these results to obtain new results on almost Gorenstein one-dimensional analytically unramified rings. [ABSTRACT FROM AUTHOR]

AbstractWe study a variant of extremal disconnectedness in frames, defined by requiring that any two disjoint open sublocales should be separable by disjoint open sublocales each of which is the interior of some zero-sublocale. In spaces, this notion was recently introduced in [1] in order to study some properties of frames of z-ideals of rings of continuous functions. The point-free approach adopted here brings to the fore some topological results not considered in [1], such as that a dense z-embedded subspace of a Tychonoff space has this property if and only if the space containing it has the property. In [1], this was proved only for the embedding X ⊆ βX. We give a ring-theoretic characterization by defining a property of f-rings in a transparent manner which shows why the property should be defined the way we do. The property is that if two annihilator ideals meet trivially, then the annihilator of each of the annihilator ideals contains a positive element such that the sum of the two elements is a non-zerodivisor. [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative ring with identity and R′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R′ is finitely generated, i.e., R′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R≤2 , then R′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integers n, m. [ABSTRACT FROM AUTHOR]

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we define the concept of homogeneous star-operations, which is a generalization of that of star-operations, and apply some properties of star-operations. Also, we define the concept of homogeneous divisor classes, which is the graded integral domain version of that of divisor classes, and study some equivalent conditions for the set of homogeneous divisor classes H D (R) of R to be a group. [ABSTRACT FROM AUTHOR]

Flatness in categories of modules over rings has been transferred to categories of acts over semigroups by Stenström in 1971. In this note we will focus on commutative cancellative semigroups with unit ("monoids") and show that this notion is useful too in considering arithmetical questions. The notion of factorable (resp. pre-Schreier) modules is extended to acts over monoids. For example it is shown that these acts can be characterized by flatness. [ABSTRACT FROM AUTHOR]

AbstractOur present work is motivated by several papers which have introduced and studied the notion of nonnil-Noetherian rings. Let A = ⊕α∈G Aα be a commutative ring with unity graded by an arbitrary commutative monoid G. We define A to be graded-nonnil-Noetherian ring if every nonnil graded ideal of A, that is, a graded ideal which is not contained in Nil(A), is finitely generated. The purpose of this paper is to give the graded version of several characterizations, properties and basic results proved in [3], [7] and [4]. We also show the non-triviality of our generalization by giving examples of graded-nonnil-Noetherian rings which are not nonnil-Noetherian and of graded-nonnil-Noetherian rings which are not graded-Noetherian. [ABSTRACT FROM AUTHOR]

Let s , a , n , d be positive integers such that n ≥ 2 and GCD (a , d) = 1 . Let P denote the defining ideal of the monomial curve associated to the sequence a , sa + d , ... , sa + nd . In this survey, we investigate the symbolic powers of P. We first show that for each i > 1 the equality P (i) = P i holds if and only if either n = 2 and a even or n = 3 , i = 2 and a ≡ 2 (mod 3) . The finitely generated property of the symbolic Rees algebra R S (P) are also explored. [ABSTRACT FROM AUTHOR]

We study polynomial closure of ideals from the point of view of star operations. We show that, if D is an integrally closed domain or a domain of residue characteristic 0, the polynomial closure of an ideal coincides with the divisorial closure. We also analyze what happens in characteristic p. [ABSTRACT FROM AUTHOR]

Given an integral domain D with quotient field K, we consider the ring Int (D) : = { f ∈ K [ X ] ; f (D) ⊆ D } of integer-valued polynomials over D. This paper deals with the question of when Int (D) is a weakly-Krull domain. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. In this paper, we introduce the concept of quasi J-ideal which is a generalization of J-ideal. A proper ideal of R is called a quasi J-ideal if its radical is a J-ideal. Many characterizations of quasi J-ideals in some special rings are obtained. We characterize rings in which every proper ideal is quasi J-ideal. Further, as a generalization of presimplifiable rings, we define the notion of quasi presimplifiable rings. We call a ring R a quasi presimplifiable ring if whenever a , b ∈ R and a = a b , then either a is a nilpotent or b is a unit. It is shown that a proper ideal I that is contained in the Jacobson radical is a quasi J-ideal (resp. J-ideal) if and only if R/I is a quasi presimplifiable (resp. presimplifiable) ring. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph with the vertex set V and let IG be its edge ideal in the polynomial ring S = K [ V ] . We compute the depth and the Castelnuovo-Mumford regularity of S/IG when G = G1 ◦ G2 or G = G1 * G2 is a graph obtained from Cohen-Macaulay bipartite graphs G1, G2 by the ◦ operation or * operation, respectively. [ABSTRACT FROM AUTHOR]

Let f : A ⟶ B be a ring homomorphism and J be an ideal of B. The subring A ⋈ f J : = { (a , f (a) + j) / a ∈ A et j ∈ J } of A × B is called the amalgamation of A with B along with J with respect to f. In this paper we investigate a general concept of ring with Noetherian spectrum, called S-Noetherian spectrum property which was introduced by Hamed, on the A ⋈ f J for a multiplicative subset S of A ⋈ f J. [ABSTRACT FROM AUTHOR]

Let B be a commutative ring with 1 ≠ 0 , 1 ≤ m < ∞ be an integer and R = B × B × ⋯ × B (m times). The extended total dot product graph TD ¯ (R) and extended zero-divisor dot product graph ZD ¯ (R) are undirected graphs with vertex sets R ∗ = R \ { (0 , 0 , ... 0) } and Z ∗ (R) = Z (R) \ { (0 , 0 , ... 0) } respectively. Two distinct vertices a = (a 1 , a 2 , ... , a m) and b = (b 1 , b 2 , ... , b m) are adjacent in TD ¯ (R) and ZD ¯ (R) if there exist positive integers k and ℓ such that a k · b ℓ = 0 with a k ≠ 0 and b ℓ ≠ 0 respectively (where a k · b ℓ = a 1 k b 1 ℓ + a 2 k b 2 ℓ + · · · + a m k b m ℓ ∈ B , denotes the normal dot product of a k and b ℓ ). In this paper, we study about connectedness, diameter and girth of TD ¯ (R) and ZD ¯ (R) . We also characterize all rings R for which TD ¯ (R) and ZD ¯ (R) are planar and outerplanar. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. The intersection graph of ideals of a ring R is an undirected simple graph denoted by Γ (R) whose vertices are in a one-to-one correspondence with non-zero proper ideals and two distinct vertices are joined by an edge if and only if the corresponding ideals of R have a non-zero intersection. Let M be a unitary R-module and let R ⋉ M be the idealization of M in R. In this paper, we investigate the interplay between the algebraic properties of R ⋉ M and the graph-theoretic properties of Γ (R ⋉ M) . Under some conditions on the ring R and the module M, we determine the exact form of ideals of R ⋉ M and characterize all rings R and modules M for which Γ (R ⋉ M) is a star graph. Also, we give a necessary and sufficient condition under which R ⋉ M is uniform and then we conclude that when Γ (R ⋉ M) is a complete graph. Among other results, some graph-theoretic properties of Γ (R ⋉ M) such as domination number, connectedness and grith are obtained. [ABSTRACT FROM AUTHOR]

We characterize the approximate Cohen–Macaulayness of a module in terms of the length function and the Hilbert coefficient of the module with respect to an almost p-standard system of parameters (a strict subclass of d-sequences). As applications, we characterize the approximate Cohen–Macaulay property of Stanley–Reisner rings, localizations, idealizations, and power series rings. Furthermore, for power series rings, we construct almost p-standard systems of parameters of them. From this result, we give a class of Cohen–Macaulay Rees algebras and give precise formulas computing all Hilbert coefficients of the formal power series ring with respect to an almost p-standard system of parameters. [ABSTRACT FROM AUTHOR]

Let D be a domain with its quotient field K. Let K [ θ ] = K [ x ] / (x n) where x is an indeterminate and θ = x + (x n) . Denote by R n = D + θ K [ θ ] , which is a subring of K [ θ ] . This paper studies several properties of Rn: coherence, weakly factorial rings, strong Mori rings, rings with Noetherian spectrum, weak normality, and t-closedness. [ABSTRACT FROM AUTHOR]

The w-operation, a "smoother" variant of the classic t-operation on the set of ideals of a commutative ring R, has recently attracted significant attention. We study when R's monoid of w-ideals is "factorial" in some sense. For example, we show that every proper w-ideal of R has a unique up to order irredundant w-factorization into w-unfactorable w-ideals if and only if R is a finite direct product of Krull domains and principal ideal rings, if and only if every w-ideal is w-quasiprincipal. We prove several analogous results regarding other kinds of "w-ideal factoriality" and generalize these results via semistar operations. [ABSTRACT FROM AUTHOR]

It is well-known that an integrally closed domain D can be expressed as the intersection of its valuation overrings but, if D is not a Prüfer domain, most of the valuation overrings of D cannot be seen as localizations of D. The Kronecker function ring of D is a classical construction of a Prüfer domain which is an overring of D[t], and its localizations at prime ideals are of the form V(t) where V runs through the valuation overrings of D. This fact can be generalized to arbitrary integral domains by expressing them as intersections of overrings which admit a unique minimal overring. In this article we first continue the study of rings admitting a unique minimal overring extending known results obtained in the 1970s and constructing examples where the integral closure is very far from being a valuation domain. Then we extend the definition of Kronecker function ring to the non-integrally closed setting by studying intersections of Nagata rings of the form A(t) for A an integral domain admitting a unique minimal overring. [ABSTRACT FROM AUTHOR]

We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey "Open Problems in Commutative Ring Theory" written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

Let A ⊆ B be an extension of integral domains, B [ [ X ] ] be the power series ring over B, and R = A + X B [ [ X ] ] be a subring of B [ [ X ] ]. In this paper, we give a complete description of v-invertible v-ideals (with nonzero trace in A) of R. We show that if B is a completely integrally closed domain and I is a fractional divisorial v-invertible ideal of R with nonzero trace over A, then I = u (J 1 + X J 2 [ [ X ] ]) for some u ∈ q f (R) , J 2 an integral divisorial v-invertible ideal of B and J 1 ⊆ J 2 a nonzero ideal of A. [ABSTRACT FROM AUTHOR]

The notion of maximal non-valuative domain is introduced and characterized. An integral domain R is called a maximal non-valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non-valuative domains have at most four maximal ideals. Various properties of maximal non-valuative domains are discussed. We characterize integrally closed maximal non-valuative domains in terms of B e ´ zout domains. A local non-integrally closed maximal non-valuative domain is also characterized. Conditions are given under which pseudo-valuation domains and maximal non-pseudo-valuation domains are maximal non-valuative domains. [ABSTRACT FROM AUTHOR]

Let A ⊆ B be two commutative rings with identity. It is well-known that if A is a Noetherian ring and B is a finitely generated A-module, then B is also a Noetherian ring. In this paper, we want to prove an analogue of the above result for SFT rings. For that, we extend the notion of SFT rings to modules. An A-module M is said to be an SFT module, if for each submodule N of M, there exist k ≥ 1 , x 1 , ... , x n ∈ N such that for each a ∈ (N : M) = { α ∈ A , α M ⊆ N } and x ∈ M , a k x ∈ ⟨ x 1 , ... , x n ⟩ . First of all, we investigate some properties of SFT modules. In fact, we show that properties of SFT rings can be generalized to SFT modules. In the end of this paper, we give a partial answer of the main question of this work. [ABSTRACT FROM AUTHOR]

Let A⊆BA\subseteq B be an extension of commutative rings with identity, (S,≤)\left(S,\le ) a nonzero strictly ordered monoid, and S*=S\{0}{S}^{* }\left=S\backslash \left\{0\right\}. Let A+〚BS*,≤〛={f∈〚BS,≤〛∣f(0)∈A}A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\}. In this study, we determine when the ring A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when SS is a strict monoid, if A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then AA is a Noetherian ring, BB is a finitely generated AA-module, and SS is finitely generated. We also show that if BB is a finitely generated AA-module over a Noetherian ring AA and (S,≤)\left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

In this note, we study the generalized fraction properties and power series properties of 풮 {\mathcal{S}} -Noetherian rings. Actually, we investigate two questions proposed in [A. Dabbabi and A. Benhissi, Generalization of the S-Noetherian concept, Arch. Math. (Brno) 59 2023, 4, 307–314]. [ABSTRACT FROM AUTHOR]

Let R be a ring and S be a multiplicative subset of R. Then R is called a uniformly S-Noetherian ring if there exists s ∈ S such that, for any ideal I of R, sI ⊆ K for some finitely generated subideal K of I. We give the Eakin-Nagata-Formanek theorem for uniformly S-Noetherian rings. In addition, the uniformly S-Noetherian properties on several ring constructions are given. The notion of u-S-injective modules is also introduced and studied. Finally, we obtain the Bass-Papp theorem for uniformly S-Noetherian rings. [ABSTRACT FROM AUTHOR]

This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups. [ABSTRACT FROM AUTHOR]

The first part of this paper introduces and studies the class of strongly nonnil-coherent rings, a subclass of the already defined and studied class of nonnil-coherent rings. Contrary to the classical result that every Noetherian ring is coherent, a nonnil-Noetherian ring need not be nonnil-coherent. To remedy this, the second part introduces and studies the class of strongly nonnil-Noetherian rings, a subclass of the class of nonnil-Noetherian rings. Some examples are also given to illustrate the results. [ABSTRACT FROM AUTHOR]