Set Of Overrings Research Articles

A finiteness condition on the set of overrings of some classes of integral domains

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.

Topological properties of semigroup primes of a commutative ring

A semigroup prime of a commutative ring R is a prime ideal of the semigroup $$(R,\cdot )$$ . One of the purposes of this paper is to study, from a topological point of view, the space $${\varvec{\mathcal {S}}}(R)$$ of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, $${\varvec{\mathcal {S}}}(R)$$ is a spectral space (after Hochster), spectral extension of $${{\mathtt {Spec}}}(R)$$ , and that the assignment $$R\mapsto {\varvec{\mathcal {S}}}(R)$$ induces a contravariant functor. We then relate—in the case R is an integral domain—the topology on $${\varvec{\mathcal {S}}}(R)$$ with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between $${\varvec{\mathcal {S}}}(R)$$ and the space $$\varvec{\mathcal {X}}(R)$$ consisting of all nonempty inverse-closed subspaces of $${{\mathtt {Spec}}}(R)$$ , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that $${\varvec{\mathcal {S}}}( R)$$ is a spectral retract of $$\varvec{\mathcal {X}}(R)$$ and we characterize when $${\varvec{\mathcal {S}}}( R)$$ is canonically homeomorphic to $$\varvec{\mathcal {X}}(R)$$ , both in general and when $${{\mathtt {Spec}}}(R)$$ is a Noetherian space. In particular, we obtain that, when R is a Bezout domain, $${\varvec{\mathcal {S}}}( R)$$ is canonically homeomorphic both to $$\varvec{\mathcal {X}}(R)$$ and to the space $$\mathtt {Overr}(R)$$ of the overrings of R (endowed with the Zariski topology). Finally, we compare the space $$\varvec{\mathcal {X}}(R)$$ with the space $${\varvec{\mathcal {S}}}(R(T))$$ of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding $${\varvec{\mathcal {X}}}(R)\hookrightarrow {\varvec{\mathcal {S}}}(R(T))$$ which makes $${\varvec{\mathcal {X}}}(R)$$ a spectral retract of $${\varvec{\mathcal {S}}}(R(T))$$ .

Graph theoretic characterizations of maximal non-valuation subrings of a field

We establish several new characterizations of maximal non-valuation subrings of a field involving several concepts of commutative algebra related to the set of prime ideals and the set of overrings. For example we show that an integral domain R of finite dimension d is a maximal non-valuation subring of a field if, and only if R is either integrally closed with a set of overrings isomorphic to a kite-graph of dimension d + 1, or is non-integrally closed with a chained set of overrings of dimension d + 1.

OVERRINGS OF THE KRONECKER FUNCTION RING Kr(D, *) OF A PRUFER *-MULTIPLICATION DOMAIN D

Let ⁄ be an e.a.b. star operation on an integrally closed do- main D, and let Kr(D,⁄) be the Kronecker function ring of D. We show that if D is a P⁄MD, then the mapping Dfi 7! Kr(Dfi,v) is a bijection from the set {Dfi} of ⁄-linked overrings of D into the set of overrings of Kr(D,v). This is a generalization of (5, Proposition 32.19) that if D is a Prufer domain, then the mapping Dfi 7! Kr(Dfi,b) is a one-to-one mapping from the set {Dfi} of overrings of D onto the set of overrings of Kr(D,b).

On the set of t-linked overrings of an integral domain

Let R be an integral domain with quotient field L. An overring T of R is t-linked over R if I−1 = R implies that (T : IT) = T for each finitely generated ideal I of R. Let Ot(R) denotes the set of all t-linked overrings of R and O(R) the set of all overrings of R. The purpose of this paper is to study some finiteness conditions on the set Ot(R). Particularly, we prove that if Ot(R) is finite, then so is O(R) and Ot(R) = O(R), and if each chain of t-linked overrings of R is finite, then each chain of overrings of R is finite. This yields that the t-linked approach is more efficient than the Gilmer’s treatment (Proc Am Math Soc 131:2337–2346, 2002). We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain, etc. We establish a connection between Ot(R) and the set of all strongly divisorial ideals of R and we conclude by a characterization of domains R that are t-linked under all their overrings.

Some finiteness conditions on the set of overrings of an integral domain

Let D be an integral domain with quotient field K and integral closure D. An overring of D is a subring of K containing D, and O(D) denotes the set of overrings of D. We consider primarily two finiteness conditions on O(D): (FO), which states that O(D) is finite, and (FC), the condition that each chain of distinct elements of O(D) is finite. (FO) is strictly stronger than (FC), but if D = D, each of (FO) and (FC) is equivalent to the condition that D is a Prufer domain with finite prime spectrum. In general D satisfies (FC) iff D satisfies (FC) and all chains of subrings of D containing D have finite length. The corresponding statement for (FO) is also valid.

The distribution of prime ideals of a Dedekind domain

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.