Narrow Results

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.¹ Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the -transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

If D ⊆ T is an extension of (commutative integral) domains and ⋆ (resp., ⋆′) is a semistar operation on D (resp., T), we define what it means for D ⊆ T to satisfy the (⋆,⋆′)-GD property. Sufficient conditions are given for (⋆,⋆′)-GD, generalizing classical sufficient conditions for GD such as flatness, openness of the contraction map of spectra and the hypotheses of the classical going-down theorem. If ⋆ is a semistar operation on a domain D, we define what it means for D to be a ⋆-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a domain, the domain extensions T of D for which is tested can be the -valuation overrings of D, the simple overrings of D, or all T. P ⋆ MDs are characterized as the -treed (resp., ) domains D which are -finite conductor domains such that is integrally closed. Several characterizations are given of the -Noetherian domains D of -dimension 1 in terms of the behavior of the (⋆,⋆′)-linked overrings of D and the ⋆-Nagata rings Na(D,⋆).

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results.

Let R ⊂ S be an extension of integral domains. We say that (R, S) is a well-centered pair of rings, if each intermediate ring T between R and S is well-centered on R, in the sense that each principal ideal of T is generated by an element of R. The aim of this paper is to study well-centered pairs of rings. We investigate the structure of the intermediate rings T between R and S that are well-centered on R. We establish the relationship between well-centered pairs and normal pairs. We present some examples and counterexamples illustrating our theory and showing the limits of our results.

Let D be an integral domain with quotient field K. For any set X, the ring Int(D^X) of integer-valued polynomials onD^X is the set of all polynomials f ∈ K[X] such that f(D^X) ⊆ D. Using the t-closure operation on fractional ideals, we find for any set X a D-algebra presentation of Int(D^X) by generators and relations for a large class of domains D, including any unique factorization domain D, and more generally any Krull domain D such that Int(D) has a regular basis, that is, a D-module basis consisting of exactly one polynomial of each degree. As a corollary we find for all such domains D an intrinsic characterization of the D-algebras that are isomorphic to a quotient of Int(D^X) for some set X. We also generalize the well-known result that a Krull domain D has a regular basis if and only if the Pólya–Ostrowski group of D (that is, the subgroup of the class group of D generated by the images of the factorial ideals of D) is trivial, if and only if the product of the height one prime ideals of finite norm q is principal for every q.

Let R ⊆ S be a unital extension of commutative rings, with the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many -subalgebras of S. Each chain of rings from R to S is finite if dim(R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim(R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

In this paper, we characterize maximal non-ACCP subrings R of a domain S in case (R, S) is a residually algebraic pair and R is semilocal. In this paper, we also consider a K-algebra S, a nonzero proper ideal I of S and a subring D of the field K and we determine necessary and sufficient conditions in order that D + I is a maximal non-ACCP subring of S. This gives an example of a maximal non-ACCP subring R of a domain S such that (R, S) is a normal pair and R is not semilocal.

Let $R\subseteq S$ 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.

Let $R\subset S$ 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.

If $A\subseteq B$ 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 $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ 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 $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. 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.

The main purpose of this paper is to answer a question which was left open in N. Jarboui and S. Aljubran [Maximal non-integrally closed subrings of an integral domain, Ric. Mat. (2020), doi.org/10.1007/s11587-020-00500-0] asking for a characterization of maximal non-integrally closed subrings of arbitrary rings.

New examples of metaideals in commutative rings are constructed. It is proved that metaideals of a commutative ring form a sublattice of the lattice of all subrings, and for any subring A of a commutative ring P, there exists the largest subring Mid_P (A) (called metaidealizer) in which A is a metaideal. Metaidealizers in several cases are described.

It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which R_M (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>2^2ℵ₀ or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.