In this survey article we discuss key open problems which could serve as a guidance for further research directions of multiplicative ideal theory and factorization theory.

Interval rings comprise a class of one-dimensional integrally closed local integral domains which are overrings of a two-dimensional regular local ring. We construct them by intersecting carefully chosen valuation rings and prove that they have various nice properties. Our interest in these rings is that they represent a major stepping stone toward classifying all integrally closed overrings of a two-dimensional regular local ring.

The reciprocal complement R(D) of an integral domain D is the subring of its fraction field generated by the reciprocals of its nonzero elements. Many properties of R(D) are determined when D is a polynomial ring in n >= 2 variables over a field. In particular, R(D) is an n-dimensional, local, non-Noetherian, non-integrally closed, non-factorial, atomic G-domain, with infinitely many prime ideals at each height other than 0 and n.

Let D be a Dedekind domain (not a field) with finite residue fields and let Int(D) be the ring of integer-valued polynomials over D. We completely classify in topological terms some relevant classes of radical unitary ideals of Int(D) (and of its overrings). This project strongly extends the classification given in a previous paper and regarding special unitary ideals, precisely the ones lying over a given maximal ideal of D. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let $D$ be a two-dimensional regular local ring. We prove there is a one-to-one correspondence between closed connected sets in the space of valuation overrings of $D$ that dominate $D$ and the integrally closed local overrings of $D$ that are not essential valuation rings or divisorial valuation rings of $D$.

Let $F$ be a field, let $D$ be a local subring of $F$, and let Val$_F(D)$ be the space of valuation rings of $F$ that dominate $D$. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of $F$ by proving that a subring $R$ of $F$ dominating $D$ is local, residually algebraic over $D$ and integrally closed in $F$ if and only if there is a closed and connected subspace $Z$ of Val$_F(D)$ such that $R$ is the intersection of the rings in $Z$. Consequently, the intersection of the rings in any closed and connected subset of Val$_F(D)$ is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian.

Let D D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of I n t ( D ) Int(D) lying over a given maximal ideal m \mathfrak m of D D . We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m \mathfrak m -adic completion of D D . Moreover, these results are naturally extended to overrings of I n t ( D ) Int(D) . As an application we provide explicit constructions of divisorial ideals of I n t ( D ) Int(D) which are not finitely generated.

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.

In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of containment of one of the principal ideals in all the others. We relate these classes to many well-studied classes of integral domains, to star operations and to classical and new ring constructions.

Recall that a ring is said to be a clean ring if every element can be expressed as the sum of a unit and an idempotent. In one variant of this definition, a ring is said to be a semi-clean ring if every element can be expressed as the sum of a unit and a periodic element. Ye's Theorem [12] states that the group ring Z(p)[C3] is semi-clean, where p is a prime integer and C3 is a cyclic group of order 3. In this article, we generalize Ye's Theorem by demonstrating that, if R is a local ring, then the group ring R[G] is semi-clean if and only if G is a torsion abelian group.

Let D be a 2-dimensional regular local ring and let Q(D) denote the quadratic tree of 2-dimensional regular local overrings of D. We explore the topology of the tree Q(D) and the family R(D) of rings obtained as intersections of rings in Q(D). If A is a finite intersection of rings in Q(D), then A is Noetherian and the structure of A is well understood. However, other rings in R(D) need not be Noetherian. The two main goals of this paper are to examine topological properties of the quadratic tree Q(D), and to examine the structure of rings in the set R(D).

We introduce an analogue of the Kronecker function ring construction in the ultrapower setting, and study when it gives a Bezout domain.

We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the "cardinal Krull dimension," is the supremum of the cardinalities of chains of prime ideals in the ring. The second, which we call the "strong cardinal Krull dimension," is a slight strengthening of the first. Our main objective is to address the following question: for which cardinal pairs (K,L) does there exist a ring of cardinality K and (strong) cardinal Krull dimension L? Relying on results from the literature, we answer this question completely in the case where K>L or K=L. We also give several constructions, utilizing valuation rings, polynomial rings, and Leavitt path algebras, of rings having cardinality K and (strong) cardinal Krull dimension L>K. The exact values of K and L that occur in this situation depend on set-theoretic assumptions.

Let (R,m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {Rn} of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {Rn}n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S=⋃n≥0Rn. Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the Rn. We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S=T∩V. If S is archimedean, then the complete integral closure S⁎ of S has the form S⁎=W∩T, where W is the rank 1 valuation overring of V.

The patch/constructible refinement of the Zariski topology on the prime spectrum of a commutative ring is well known and well studied. Recently, Fontana and Loper gave an equivalent definition of this topology using ultrafilters. In this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring. We study various properties of these topologies. As applications we use the ultrafilters to classify all the compact subsets of a spectral space in the Zariski topology and we classify Grothendieck's retrocompact spaces again using ultrafilters.

Define the nth real cyclotomic polynomial to be the minimal polynomial over Z of ζn+ζn−1, where ζn=e2πi/n is a primitive nth root of unity. We prove that the real cyclotomic polynomials can be formed from compositions of polynomials closely related to the Chebyshev polynomials of the first kind. We use these relations to determine the resultant of two real cyclotomic polynomials.

Let K be a field with rank one valuation and V the valuation domain of K. For a subset E of V, the ring of integer-valued polynomials on E isInt (E, V) = {f is an element of K [x] vertical bar f (E) subset of V}.A question of interest regarding Int (E,V) is: for which E is Int (E, V) a Prufer domain? In this paper, we contribute a partial answer to this question. We classify exactly when Int (E,V) is Prufer in the case where the elements of E comprise a pseudo-convergent sequence in V. Our work expands on earlier results that apply when V is a discrete valuation domain.

The integer split quaternions form a noncommutative algebra over Z. We describe the prime and maximal spectrum of the integer split quaternions and investigate integer-valued polynomials over this ring. We prove that the set of such polynomials forms a ring, and proceed to study its prime and maximal ideals. In particular we completely classify the primes above 0, we obtain partial characterizations of primes above odd prime integers, and we give sufficient conditions for building maximal ideals above 2.

We prove that the theory of root closed monoids is axiomatizable, but not finitely axiomatizable. Some directions for further research are presented.

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains.