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In this paper, we deal with integral domains with only a few semistar operations. Mainly, we give complete characterizations of integral domains with exactly three (respectively, four) semistar operations (respectively, semistar operation of finite character). Examples to illustrate our main theorems are given. [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 D be a domain and let Int (D) and In t R (D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int (Z) with all the value ideals of I not being proper, then I = Int (Z). This is known as the Skolem property, which does not hold in Z [ x ]. One obstruction to Int (D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on In t R (D). We determine that the Skolem property on In t R (D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion. [ABSTRACT FROM AUTHOR]

Over Prüfer domains, we characterize idempotent by nilpotent 2-products of 2 × 2 matrices. Nilpotents are always such products. We also provide large classes of rings over which every 2 × 2 idempotent matrix is such a product. Finally, for 2 × 2 matrices over GCD domains, idempotent–nilpotent products which are also nilpotent–idempotent products are characterized. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited. [ABSTRACT FROM AUTHOR]

We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D. Design/methodology/approach – The methods used are from the topic multiplicative ideal theory from commutative ring theory. Findings – Let S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D. Originality/value – Motivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

This paper contributes to the study of the core of an ideal in integral domains. Our aim is to develop explicit formulas for the core in various classes of Prüfer domains. We pay particular attention to relevant ideal-theoretic notions such as stability, invertibility, and h -local property. We also provide decomposition results for the core of an ideal in integral domains with effectual ramifications to Prüfer domains. All main results are illustrated with original examples, where we explicitly compute the core. We also provide counter-examples to test the limits of the assumptions used in the main results. [ABSTRACT FROM AUTHOR]

Let R ⊂ S be an extension of integral domains. The domain R is said to be a maximal non-integrally closed subring of S if R is not integrally closed in S, while each subring of S properly containing R is integrally closed in S. Jaballah (J Algebra Appl 11(5):1250041, 18pp, 2012) has characterized these domains when S is the quotient field of R. The main purpose of this paper is to study this kind of ring extensions in the general case. Some examples are provided to illustrate our obtained results. Our main result also answers a key question raised by Gilmer and Heinzer (J Math Kyoto Univ 7(2):133–150, 1967). [ABSTRACT FROM AUTHOR]

Let R be a domain. It is proved that if wG-gldim (R) < ∞ , then the class of finitely generated G -flat modules and the class of finitely generated G -projective modules coincide. It is also proved that an integrally closed domain R is a Prüfer domain if and only if FP − id R (R) ≤ 1 , if and only if wG-gldim (R) ≤ 1. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, X be an indeterminate and I c (R) be the set of elements a of R such that there exists an R -homomorphism of rings σ : R [ [ X ] ] → R with σ (X) = a. O'Malley called R to be power invariant (respectively, strongly power invariant) if whenever S is a ring such that R [ [ X ] ] is isomorphic to S [ [ X ] ] (respectively, whenever S is a ring and φ is an isomorphism of R [ [ X ] ] onto S [ [ X ] ]), then R and S are isomorphic (respectively, then there exists an S -automorphism ψ of S [ [ X ] ] such that ψ (X) = φ (X)) [M. O'Malley, Isomorphic power series rings, Pacific J. Math. 41(2) (1972) 503–512]. We prove that a ring R is power invariant in each of the following case: (1) R is a domain in which I c (R) is comparable to each radical ideal of R (for instance a domain with Krull dimension one), (2) R is a domain in which Jac (R) (i.e. the Jacobson radical of R) is comparable to each radical ideal of R and (3) R is a Prüfer domain. Also in each of the aforementioned case, we prove that either R is strongly power invariant or R is isomorphic to a quasi-local power series ring. Let M be a unital module over R. We show that if R is reduced and strongly power invariant, then Nagata's idealization ring R (+) M is strongly power invariant (but the converse is false). Ishibashi called a ring R to be strongly n -power invariant if whenever S is a ring and φ is an isomorphism of R [ [ X 1 , ... , X n ] ] onto S [ [ X 1 , ... , X n ] ] , then there exists an S -automorphism ψ of S [ [ X 1 , ... , X n ] ] such that ψ (X i) = φ (X i) for each i. We prove that if R is a ring in which I c (R) is nil, then R is strongly n -power invariant for all positive integer n. We deduce that every polynomial ring in finitely many indeterminates is strongly n -power invariant for all positive integer n. [ABSTRACT FROM AUTHOR]

Let D be a pseudo-valuation domain with associated valuation domain V and let E be a nonempty subset of V. We show that Int (E , D) is a globalized pseudo-valuation domain if and only if Int (E , V) is a Prüfer domain. In this case, Int (E , V) is the associated Prüfer domain of Int (E , D) ; Int R (E , D) is a globalized pseudo-valuation domain with associated Prüfer domain Int R (E , V) ; furthermore, every ring between Int (E , D) and Int R (E , D) is a globalized pseudo-valuation domain. Also, in this case, we describe the unitary maximal ideals of Int (E , V) and show that Int R (E , V) is a Bézout domain. [ABSTRACT FROM AUTHOR]

In this paper we present conditions under which a Prüfer domain has a maximal subring. We study the Prüfer and Bézout properties which are shared between an integral domain and its maximal subrings too. [ABSTRACT FROM AUTHOR]

If A ⊆ B are (commutative) rings, [ A , B ] denotes the set of intermediate rings and (A , B) is called an almost valuation (AV)-ring pair if each element of [ A , B ] is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let R ⊆ S be rings, with R ¯ S denoting the integral closure of R in S. Then (R , S) is an AV-ring pair if and only if both (R , R ¯ S) and ( R ¯ S , S) are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions R ⊆ S. If (R , S) is an AV-ring pair, then R ⊆ S is a P-extension. The AV-ring pairs (R , S) arising from root extensions are studied extensively. Transfer results for the "AV-ring" property are obtained for pullbacks of (B , I , D) type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for (R , S) being an AV-ring pair to entail that S is an overring of R , but there exist domain-theoretic counter-examples to such a conclusion in general. If (R , S) is an AV-ring pair and R ⊆ S satisfies FCP, then each intermediate ring either contains or is contained in R ¯ S . While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain. [ABSTRACT FROM AUTHOR]

We explore the connection between atomicity in Prüfer domains and their corresponding class groups. We observe that a class group of infinite order is necessary for non-Noetherian almost Dedekind and Prüfer domains of finite character to be atomic. We construct a non-Noetherian almost Dedekind domain and exhibit a generating set for the ideal class semigroup. [ABSTRACT FROM AUTHOR]

Let V be a valuation domain and let E be a subset of V. For a rank-one valuation domain V , there is a characterization of when Int (E , V) is a Prüfer domain. For a general valuation domain V , we show that Int (E , V) is a Prüfer domain if and only if E is precompact, or there exists a rank-one prime ideal P of V and Int (E , V P) is a Prüfer domain. Then we show that the following statements are equivalent: (1) Int (E , V) is a Prüfer domain; (2) it has the strong 2-generator property; (3) it has the almost strong Skolem property. In this case, by showing that Int (E , V) is almost local-global, we obtain that it has the stacked bases property and the Steinitz property. For a Prüfer domain D , we show that the following statements are equivalent: (1) Int (D) is a Prüfer domain; (2) it has the 2-generator property; (3) it has the almost strong Skolem property. In this case, Int (D) is not necessarily almost local-global, but we show that it has the Steinitz property. [ABSTRACT FROM AUTHOR]

In this paper we classify commutative ring extensions with exactly two non Prüfer domain intermediate rings. An initial step involves the description of the commutative ring extensions with only one non Prüfer domain intermediate ring. Some generalizations to the context of rings with zero divisors are proved. We also answer a question which was left open in Jarboui and Aljubran (Ric Mat, , 2020). More precisely, let S = K [ y 1 ] ]... [ y t ] ] be a K-algebra (not necessarily finitely generated over the field K) having Krull dimension n ≥ 1 . Let I be a nonzero proper ideal of S (not necessarily maximal in S) and D be a proper subring of K. We provide necessary and sufficient conditions in order that R = D + I is a maximal non-integrally closed subring of S. [ABSTRACT FROM AUTHOR]

Let R ⊂ S be an extension of integral domains. The ring R is said to be maximal non-Prüfer subring of S if R is not a Prüfer domain, while each subring of S properly containing R is a Prüfer domain. Jaballah has characterized this kind of ring extensions in case S is a field [A. Jaballah, Maximal non-Prüfer and maximal non-integrally closed subrings of a field, J. Algebra Appl.11(5) (2012) 1250041, 18 pp.]. The aim of this paper is to deal with the case where S is any integral domain which is not necessarily a field. Several examples are provided to illustrate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the linkage of ideals, in Noetherian and non-Noetherian settings, with the aim to establish new characterizations of classical notions of domains through linkage theory. Two main results assert that a Noetherian domain is Dedekind if and only if it has the primary linkage property; and a domain is almost Dedekind (resp., Prüfer) if and only if it has the linkage (resp., finite linkage) property. Also, we prove that a finite-dimensional valuation domain is a DVR (i.e., Noetherian) if and only if it has the primary linkage property. [ABSTRACT FROM AUTHOR]

This paper deals with minimal reductions and core of ideals in various settings of pullback constructions with the aim of building original examples, where we explicitly compute the core. To this purpose, we use techniques and objects from multiplicative ideal theory to investigate the existence of minimal reductions in Section 2 and then develop explicit formulas for the core in Section 3. The last section features illustrative examples and counterexamples. [ABSTRACT FROM AUTHOR]

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prüfer domains. In the present paper, we investigate some analogs of these concepts for modules over group rings. [ABSTRACT FROM AUTHOR]

We study a dimensional-invariant related in some way to the conditions of Krull-Akizuki's theorem. [ABSTRACT FROM AUTHOR]

This paper deals with reductions of ideals in various settings of pullback constructions. Precisely, we investigate reductions of several types of ideals in both generic and classical pullbacks. We also characterize pullbacks where reductions of a class of ideals extend to reductions of their respective extended ideals. All results are illustrated with original examples. [ABSTRACT FROM AUTHOR]

Let R ⊆ S be an extension of integral domains with only finitely many intermediate rings, where R is not a field and S is not necessarily the quotient field of R or R is not necessarily integrally closed in S. In this paper, we exactly determine the number of intermediate rings between R and S and give a way to compute it. [ABSTRACT FROM AUTHOR]

Prüfer domains and subclasses of integral domains such as Dedekind domains admit characterizations by means of the properties of their ideal lattices. Interestingly, a Leavitt path algebra L , in spite of being noncommutative and possessing plenty of zero divisors, seems to have its ideal lattices possess the characterizing properties of these special domains. In [The multiplicative ideal theory of Leavitt path algebras, J. Algebra 487 (2017) 173–199], it was shown that the ideals of L satisfy the distributive law, a property of Prüfer domains and that L is a multiplication ring, a property of Dedekind domains. In this paper, we first show that L satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers a , b , c , gcd (a , b) ⋅ lcm (a , b) = a ⋅ b and a ⋅ gcd (b , c) = gcd (a b , a c). We also show that L satisfies a characterizing property of almost Dedekind domains in terms of the ideals whose radicals are prime ideals. Finally, we give necessary and sufficient conditions under which L satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals. [ABSTRACT FROM AUTHOR]

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if R ⊆ S are rings and D = (d ij) is an n × n matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of n × n matrices with entries in R if and only if each d ij is integral over R. If R ⊆ S are rings with corresponding full rings of n × n matrices R n and S n , then (R n , S n) is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair (Λ , Γ) of noncommutative (in fact, full matrix) rings such that Λ ⊂ Γ is (resp., is not) a minimal ring extension; it can be further arranged that (Λ , Γ) is a normal pair or that Λ ⊂ Γ is an integral extension. [ABSTRACT FROM AUTHOR]

To study the question of whether every two-dimensional Prüfer domain possesses the stacked bases property, we consider the particular case of the Prüfer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and in this case we reduce the study of the stacked bases property to questions concerning 2 × 2 -matrices. [ABSTRACT FROM AUTHOR]

We introduce the notion of ultra star operation on ultraproduct of integral domains as a map from the set of induced ideals into the set of induced ideals satisfying the traditional properties of star operations. A case of special interest is the construction of an ultra star operation on the ultraproduct of integral domains Ri's from some given star operations *i on Ri's. We provide the ultra b-operation and the ultra v-operation. Given an arbitrary star operation * on the ultraproduct of some integral domains, we pose the problem of whether the restriction of * to the set of induced ideals is necessarily an ultra star operation. We show that the ultraproduct of integral domains Ri's is a *-Prufer domain if and only if Ri is a *i-Prufer domain for U-many i. [ABSTRACT FROM AUTHOR]

Let D be an integral domain and a star operation defined on D. We say that D is a -power conductor domain (-PCD) if for each pair and for each positive integer n we have We study -PCDs and characterize them as root closed domains satisfying for all nonzero a, b and all natural numbers. From this it follows easily that Prüfer domains are d-PCDs (where d denotes the trivial star operation), and v-domains (e.g. Krull domains) are v-PCDs. We also consider when a -PCD is completely integrally closed, and this leads to new characterizations of Krull domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a w-PCD. [ABSTRACT FROM AUTHOR]

The order dual [ Fil R R ] du of the set Fil R R of all right topologizing filters on a fixed but arbitrary ring R is a complete lattice ordered monoid with respect to the (order dual) of inclusion and a monoid operation ' : ' that is, in general, noncommutative. It is known that [ Fil R R ] du is always left residuated, meaning, for each pair F , G ∈ Fil R R there exists a smallest H ∈ Fil R R such that H : G ⊇ F , but is not, in general, right residuated (there exists a smallest H such that G : H ⊇ F ). Rings R for which [ Fil R R ] du is both left and right residuated are shown to satisfy the DCC on left annihilator ideals and possess only finitely many minimal prime ideals. It is shown that every maximal ideal P of a commutative ring R gives rise to an onto homomorphism of lattice ordered monoids φ ^ P from [ Fil R ] du to [ Fil R P ] du where RP denotes the localization of R at P. The kernel ≡ φ ^ P of φ ^ P is a congruence on [ Fil R ] du whose properties we explore. Defining Rad (Fil R) to be the intersection of all congruences ≡ φ ^ P as P ranges through all maximal ideals of R, we show that for commutative VNR rings R, Rad (Fil R) is trivial (the identity congruence) precisely if R is noetherian (and thus a finite product of fields). It is shown further that for arbitrary commutative rings R, Rad (Fil R) is trivial whenever Fil R is commutative (meaning, the monoid operation ' : ' on Fil R is commutative). This yields, for such rings R, a subdirect embedding of [ Fil R ] du into the product of all [ Fil R P ] du as P ranges through all maximal ideals of R. The theory developed is used to prove that a Prüfer domain R for which Fil R is commutative, is necessarily Dedekind. [ABSTRACT FROM AUTHOR]

Let be a semistar operation on a domain D, the finite-type semistar operation associated to , and D a Prüfer -multiplication domain (P MD). For the special case of a Prüfer domain (where is equal to the identity semistar operation), we show that a nonzero prime P of D is sharp, that is, that , where the intersection is taken over the maximal ideals M of D that do not contain P, if and only if two closely related spectral semistar operations on D differ. We then give an appropriate definition of -sharpness for an arbitrary P MD D and show that a nonzero prime P of D is -sharp if and only if its extension to the -Nagata ring of D is sharp. Calling a P MD -sharp (-doublesharp) if each maximal (prime) -ideal of D is sharp, we also prove that such a D is -doublesharp if and only if each -linked overring of D is -sharp. [ABSTRACT FROM AUTHOR]

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary. [ABSTRACT FROM AUTHOR]

A commutative ring R is said to be maximal non-prime ideally equal subring of S, if Spec(R) ≠ Spec(S), whereas Spec(T) = Spec(S) for any subring T of S properly containing R. The aim of this paper is to give a complete characterization of this class of rings. [ABSTRACT FROM AUTHOR]

We propose to give a positive answer to the following question: is R⟨X,Y⟩ strong S when R⟨X⟩ is strong S? in case R is obtained by a (T, I, D) construction. [ABSTRACT FROM AUTHOR]

Let V be a rank one valuation domain with quotient field K . We characterize the subsets S of V for which the ring Int ( S , V ) = { f ∈ K [ X ] | f ( S ) ⊆ V } of integer-valued polynomials over S is a Prüfer domain. The characterization is obtained by means of the notion of pseudo-monotone sequence and pseudo-limit in the sense of Chabert, which generalize the classical notions of pseudo-convergent sequence and pseudo-limit by Ostrowski and Kaplansky, respectively. We show that Int ( S , V ) is Prüfer if and only if no element of the algebraic closure K ‾ of K is a pseudo-limit of a pseudo-monotone sequence contained in S , with respect to some extension of V to K ‾ . This result expands a recent result by Loper and Werner. [ABSTRACT FROM AUTHOR]

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R⟨X⟩. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38] related to the transfer of Prüfer conditions to these two constructions. [ABSTRACT FROM AUTHOR]

As an extension of the class of Dedekind domains, we have introduced and studied the class of multiplicatively pinched-Dedekind domains (MPD domains) and the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains) ([T. Dumitrescu and S. U. Rahman, A class of pinched domains, Bull. Math. Soc. Sci. Math. Roumanie 52 (2009) 41-55] and [T. Dumitrescu and S. U. Rahman, A class of pinched domains II, Comm. Algebra 39 (2011) 1394-1403]). The main interest of this paper is to study GMPD domains that have only finitely many overrings. [ABSTRACT FROM AUTHOR]

In this paper, we reinvestigate the notion of fuzzy semistar operation introduced in [5] . We show how a fuzzy semistar operation on an integral domain R induces canonically a fuzzy semistar operation on an overring T of R and conversely how a fuzzy semistar operation on T induces canonically a semistar operation on R . As an application, new characterizations of Prüfer domains and complete description of the set of all fuzzy semistar operations of finite character on Prüfer domains are obtained. We also characterize conducive domains. [ABSTRACT FROM AUTHOR]

In this paper we show that, if R is a ring and σ an endomorphism of R, then the skew Hurwitz series ring T = (HR, σ ) is an n-clean ring if and only if R is an n-clean ring. Moreover, if R is an integral domain and a torsion-free Z-module, then T = (HR, σ ) is a Prufer domain if and only if R is a field. Also, we investigate when the ring T = (HR, σ ) is g(x)-clean, (n, g(x))-clean and a Neat ring.

Let F be a field, and let Zar ( F ) be the space of valuation rings of F with respect to the Zariski topology. We prove that if X is a quasicompact set of rank one valuation rings in Zar ( F ) whose maximal ideals do not intersect to 0, then the intersection of the rings in X is an integral domain with quotient field F such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to 0, and (b) one-dimensional Prüfer domains with nonzero Jacobson radical and quotient field F . The necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to 0 is motivated by settings in which the valuation rings considered all dominate a given local ring. [ABSTRACT FROM AUTHOR]