Mori Domains Research Articles

N-absorbing ideal factorization in commutative rings

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form A + XB [ X ] where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some n ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given.

Generalized transforms of graded Strong Mori domains

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we show that if R is a graded Strong Mori domain, then every homogeneously t-linked overring of R contained in R hwg is a graded Strong Mori domain, where Rhwg is the homogeneous w-global transform of R; and if R is a graded Strong Mori domain, then R S is also a graded Strong Mori domain, where S is a multiplicative set of ideals of R generated by homogeneous ideals of R.

Unique decompositions into w-ideals for strong Mori domains

A commutative ring [Formula: see text] has the unique decomposition into ideals (UDI) property if, for any [Formula: see text]-module that decomposes into a finite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism classes of the indecomposable ideals. In [P. Goeters and B. Olberding, Unique decomposition into ideals for Noetherian domains, J. Pure Appl. Algebra 165 (2001) 169–182], the UDI property has been characterized for Noetherian integral domains. In this paper, we aim to study the UDI-like property for strong Mori domains; domains satisfying the ascending chain condition on [Formula: see text]-ideals.

Ring–theoretic properties of locally Mori domains and rings of the form $$\mathrm {Int}(E,D)$$

Ring–theoretic properties of locally Mori domains and rings of the form $$\mathrm {Int}(E,D)$$

On some classes of integral domains with finitely many star operations of finite type

Let R be an integral domain. We investigate when R has only finitely many star operations of finite type. We start by investigating some classes of integral domains with at most two star operations of finite type. Our aim is to generalize or find analogues of several characterizations of well-known classes of integral domains with at most two star operations. Next we concentrate on Mori domains with only finitely many star operations of finite type in a similar study done for Noetherian domains. We end by a section on pullbacks to enrich the literature with new and original examples and also to illustrate several results in the paper.

Some results on t–locally Strong Mori domains and their rings of integer–valued polynomials

An integral domain D is said to be t–locally Strong Mori (for short, t–LSM) if $$D_\mathfrak {m}$$ is Strong Mori for all t–maximal ideals $$\mathfrak {m}$$ of D. This paper studies some ring–theoretic properties of t–LSM domains and the algebra structure of rings of integer–valued polynomials arising from t–LSM domains. Among other things, we investigate the property of being a t–LSM domain in the t–flat overring extension, the t–Nagata ring, the polynomial ring, pullback construction and the power series ring. Also, we study $${\mathrm {Int}}(D)$$ over a t–LSM domain D. Precisely, we are interested in the Krull dimension, the trivial case and some module structure properties.

On strongly primary monoids and domains

A commutative integral domain is primary if and only if it is one-dimensional and local. A domain is strongly primary if and only if it is local and each nonzero principal ideal contains a power of the maximal ideal. Hence, one-dimensional local Mori domains are strongly primary. We prove among other results that if R is a domain such that the conductor vanishes, then is finite; that is, there exists a positive integer k such that each nonzero nonunit of R is a product of at most k irreducible elements. Using this result, we obtain that every strongly primary domain is locally tame, and that a domain R is globally tame if and only if In particular, we answer Problem 38 of the open problem list by Cahen et al. in the affirmative. Many of our results are formulated for monoids.

Trace properties and the rings $$R({\scriptstyle {\mathrm{X}}})$$ and $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$

An integral domain R is an RTP domain (or has the radical trace property) (resp. an LTP domain), if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp. $$I(R:I)R_P=PR_P$$ for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study when the Nagata ring $$R({\scriptstyle {\mathrm{X}}})$$ and the ring $$R\langle {\scriptstyle {\mathrm{X}}}\rangle $$ are LTP (resp. RTP) domains in different contexts of integral domains such as integrally closed domains, Noetherian and Mori domains, pseudo-valuation domains and more. We also study the descent of these notions from particular overrings of R to R itself.

On S-Mori domains

In [A. Hamed and S. Hizem, Modules satisfying the [Formula: see text]-Neotherian property and [Formula: see text]-ACCR, Comm. Algebra 44 (2016) 1941–1951], the authors introduced the notion of [Formula: see text]-stationary sequence, which is a generalization of the ascending chain condition (ACC) property. Let [Formula: see text] be an integral domain and [Formula: see text] a multiplicative subset of [Formula: see text] An increasing sequence [Formula: see text] of ideals of [Formula: see text] is called [Formula: see text]-stationary if there exist a positive integer [Formula: see text] and an [Formula: see text] such that for each [Formula: see text] [Formula: see text] As a generalization of Mori domains, we define [Formula: see text] to be an [Formula: see text]-Mori domain if every increasing sequence of integral divisorial ideals of [Formula: see text] is [Formula: see text]-stationary. In this paper, we investigate the class of such domains [Formula: see text] and we give an [Formula: see text]-version of classical results about Mori domains.

Factoring ideals and stability in integral domains

In an integral domain $R$, a nonzero ideal is called a \textit {weakly $ES$-stable ideal} if it can be factored into a product of an invertible ideal and an idempotent ideal of $R$; and $R$ is called a \textit {weakly $ES$-stable domain} if every nonzero ideal is a weakly $ES$-stable ideal. This paper studies the notion of weakly $ES$-stability in various contexts of integral domains such as Noetherian and Mori domains, valuation and Pr\ufer domains, pullbacks and more. In particular, we establish strong connections between this notion and well-known stability conditions, namely, Lipman, Sally-Vasconcelos and Eakin-Sathaye stabilities.

Radical perfectness of prime ideals in certain integral domains

For a UMT-domain $D$, we characterize when the polynomial ring $D[X]$ is $t$-compactly packed and every prime $t$-ideal of $D[X]$ is radically perfect. As a corollary, for a quasi-Pr\"ufer domain $D$, we also characterize when every prime ideal of $D[X]$ is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.

A HALF-CENTERED STAR-OPERATION ON AN INTEGRAL DOMAIN

In this paper, we study the natural star-operation defined by the set of associated primes of principal ideals of an integral domain, which is called the g-operation. We are mainly concerned with the ideal-theoretic properties of this star-operation. In particular, we investigate DG-domains (i.e., integral domains in which each ideal is a g-ideal), which form a proper subclass of the DW-domains. In order to provide some original examples, we examine the transfer of the DG-property to pullbacks. As an application of the g-operation, it is shown that w-divisorial Mori domains can be seen as a Gorenstein analogue of Krull domains.

On S-strong Mori domains

Let D be an integral domain, S be a (not necessarily saturated) multiplicative subset of D, w be the so-called w-operation on D, and M be a unitary D-module. As generalizations of strong Mori domains (respectively, UFDs) and strong Mori modules, we define D to be an S-strong Mori domain (respectively, S-factorial domain) if for each nonzero ideal I of D, there exist an s∈S and a w-finite type (respectively, principal) ideal J of D such that sI⊆J⊆Iw; and M to be an S-strong Mori module if M is a w-module and for each nonzero submodule N of M, there exist an s∈S and a w-finite type submodule F of N such that sN⊆F⊆Nw. This paper presents some properties of S-strong Mori domains, S-factorial domains and S-strong Mori modules.

Trace Properties and Integral Domains, II

An integral domain R has the radical trace property, or is an RTP domain, if I(R: I) is a radical ideal for each nonzero noninvertible ideal I. A related property is LTP: I(R: I)R P = PR P for each minimal prime P of I(R: I). It is clear that each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. The ideal I(R: I) is the trace of I and I is a trace ideal if I(R: I) = I. A new characterization for RTP domains is established for Noetherian domains, Mori domains, and Prüfer domains. In these special cases, R is an RTP domain if and only if IB(R: IB) = I for each trace ideal I and each ideal B of (R: I) that contains I (here (R: I) is a ring as I is a trace ideal). In the Prüfer case, having IJ(R: IJ) = I for each trace ideal I and each ideal J of R containing I is enough to have the radical trace property. On the other hand, an example is given of a one-dimensional local Noetherian domain R that is not an RTP domain but does satisfy IJ(R: IJ) = I for each trace ideal I and each ideal J of R that contains I.

Star Stability and Star Regularity for Mori Domains

In the last few years, the concepts of stability and Clifford regularity have been fruitful extended by using star operations. In this paper we study and put in relation these properties for Noetherian and Mori domains, substantially improving several results present in the literature.

Integer-Valued Polynomial Rings, t-Closure, and Associated Primes

Given an integral domain D with quotient field K, the ring of integer-valued polynomials on D is the subring {f(X) ∈ K[X]: f(D) ⊂ D} of the polynomial ring K[X]. Using the tools of t-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, Prüfer v-multiplication domains, and Mori domains.

On some classes of integral domains defined by Krullʼs a.b. operations

Let D be an integral domain with quotient field K. The b-operation that associates to each nonzero D-submodule E of K, E b : = ⋂ { E V | V valuation overring of D } , is a semistar operation that plays an important role in many questions of ring theory (e.g., if I is a nonzero ideal in D, I b coincides with its integral closure). In a first part of the paper, we study the integral domains that are b-Noetherian (i.e., such that, for each nonzero ideal I of D, I b = J b for some a finitely generated ideal J of D). For instance, we prove that a b-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily b-Noetherian. We also characterize several distinguished classes of b-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type ⋆ a canonically associated to a given semistar operation ⋆ (for instance, the b-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations d, t, w and v equal to some of the canonically associated e.a.b. operations b, t a , w a and v a .

Some results on a conjecture regarding Mori domain

Based on the famous Mori-Nagata Theorem: The integral closure of a noetherian domain is a Krull domain, similar assertion was conjectured for Mori domain as follows: The complete integral closure of a Mori domain is a Krull domain. The conjecture is positive for a noetherian domain, Krull domain, a semi normal Mori domain [6] and Mori domains for which (D : D*) ≠ 0. In general, as M. Roitman has noted [26], the conjecture is not true. In this paper, an attempt is being made, among other things, to prove that the conjecture is true for a one dimensional Mori domain and for a finite dimensional AV- Mori domain. On the other hand, using the idea of conductor ideals, a simplified proof is given that the conjecture is true for semi normal Mori domains with nonzero pseudo radical.

On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b ∈ H , let ω ( H , b ) be the smallest N ∈ N 0 ∪ { ∞ } having the following property: if n ∈ N and a 1 , … , a n ∈ H are such that b divides a 1 ⋅ … ⋅ a n , then b already divides a subproduct of a 1 ⋅ … ⋅ a n consisting of at most N factors. The monoid H is called tame if sup { ω ( H , u ) ∣ u is an atom of H } < ∞ . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M .

Condensed and Strongly Condensed Domains

AbstractThis paper deals with the concepts of condensed and strongly condensed domains. By definition, an integral domain R is condensed (resp. strongly condensed) if each pair of ideals I and J of R, I J = ﹛ab/a ∈ I, b ∈ J﹜ (resp. I J = a J for some a ∈ I or I J = Ib for some b ∈ J). More precisely, we investigate the ideal theory of condensed and strongly condensed domains in Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks.