Multiplicative Ideal Theory Research Articles

Adjacency-like conditions and induced ideal graphs

In this paper we examine some natural ideal conditions and show how graphs can be defined that give a visualization of these conditions. We examine the interplay between the multiplicative ideal theory and the graph theoretic structure of the associated graph. Behavior of these graphs under standard ring extensions are studied, and in conjunction with the theory, some classical results and connections are made.

Some examples in Gorenstein multiplicative ideal theory

In this paper, we construct some non-integrally closed domains in Gorenstein multiplicative ideal theory. For example, we show that there exists a Gorenstein Prüfer domain which is neither Gorenstein Dedekind nor Prüfer, and there exists a Gorenstein Krull domain which is neither Gorenstein Dedekind nor Krull. Also, we construct a non-integrally closed non-coherent domain in which all Gorenstein projective (resp., injective, flat) modules are projective (resp., injective, flat).

When is (D,K) anS-accr pair?

PurposeThe purpose of this article is to determine necessary and sufficient conditions in order that (D,K) to be anS-accr pair, whereDis an integral domain andKis a field which containsDas a subring andSis a multiplicatively closed subset ofD.Design/methodology/approachThe methods used are from the topic multiplicative ideal theory from commutative ring theory.FindingsLetSbe a strongly multiplicatively closed subset of an integral domainDsuch that the ring of fractions ofDwith respect toSis not a field. Then it is shown that (D,K) is anS-accr pair if and only ifKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSinKis a one-dimensional Prüfer domain. LetD,S,Kbe as above. If each intermediate domain betweenDandKsatisfiesS-strong accr*, then it is shown thatKis algebraic overDand the integral closure of the ring of fractions ofDwith respect toSis a Dedekind domain; the separable degree ofKoverFis finite andKhas finite exponent overF, whereFis the quotient field ofD.Originality/valueMotivated by the work of some researchers onS-accr, the concept ofS-strong accr* is introduced and we determine some necessary conditions in order that (D,K) to be anS-strong accr* pair. This study helps us to understand the behaviour of the rings betweenDandK.

FT-domains and Gorenstein Prüfer v-multiplication domains

It was shown by Kang (1989) that a domain R is a Krull domain if and only if R is a Mori domain and a PvMD. In this paper, we extend this result to Gorenstein multiplicative ideal theory. To do this, we introduce the concepts of FT-domains and G-PvMDs, and study them by a new star-operation, i.e., the f-operation. We prove that (1) a domain R is an integrally closed FT-domain if and only if R is a P-domain; (2) a domain R is a G-PvMD if and only if R is a g-coherent FT-domain; (3) a domain R is a G-Krull domain if and only if R is a Mori domain and a G-Pv$v$MD.

Products and intersections of prime-power ideals in Leavitt path algebras

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra [Formula: see text]. Specifically, we show that factorizations of an ideal in [Formula: see text] into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in [Formula: see text], which turn out to be prime-power ideals of a special type, as well as ideals that can be factored into products or intersections of finitely many completely irreducible ideals.

Multiplicative Ideal Theory of Semigroups

Multiplicative Ideal Theory of Semigroups

Minimal reductions and core of ideals in pullbacks

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.

Core of ideals in one-dimensional Noetherian domains

This paper contributes to the study of the core of ideals in Noetherian rings. We use techniques and objects from multiplicative ideal theory to develop explicit formulas for the core in various classes of one-dimensional Noetherian domains. All results are illustrated with original examples, where we explicitly compute the core.

On Prüfer-like properties of Leavitt path algebras

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 [Formula: see text], 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 [Formula: see text] satisfy the distributive law, a property of Prüfer domains and that [Formula: see text] is a multiplication ring, a property of Dedekind domains. In this paper, we first show that [Formula: see text] satisfies two more characterizing properties of Prüfer domains which are the ideal versions of two theorems in Elementary Number Theory, namely, for positive integers [Formula: see text], [Formula: see text] and [Formula: see text]. We also show that [Formula: see text] 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 [Formula: see text] satisfies another important characterizing property of almost Dedekind domains, namely, the cancellative property of its nonzero ideals.

The monotone catenary degree of monoids of ideals

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper, we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of [Formula: see text]-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions, we establish arithmetical finiteness results, in particular, for the monotone catenary degree and for the structure of sets of lengths and of their unions.

The multiplicative ideal theory of Leavitt path algebras

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the distributive law A∩(B+C)=(A∩B)+(A∩C) holds for any three two-sided ideals of L. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A⊆B, there is always an ideal C such that A=BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. The irreducible ideals of L turn out to be precisely the primary ideals of L. It is shown that an ideal I of L is a product of finitely many prime ideals if and only the graded part gr(I) of I is a product of prime ideals and that I/gr(I) is finitely generated with a generating set of cardinality no more than the number of distinct prime ideals in the prime factorization of gr(I). As an application, it is shown that if E is a finite graph, then every ideal of L is a product of prime ideals. The same conclusion holds if L is two-sided artinian or two-sided noetherian. Examples are constructed verifying whether some of the well-known theorems in the ideal theory of commutative rings such as the Cohen's theorem on prime ideals and the characterizing theorem on ZPI rings hold for Leavitt path algebras.

Fuzzy Logic versus Classical Logic: An Example in Multiplicative Ideal Theory

We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classical logic to fuzzy logic. Precisely, we present an example to show that, in the fuzzy context, the fact that the supremum is naturally used in lieu of the union can alter an argument that may work in the classical context.

Core of ideals in integral domains

This paper uses objects and techniques from multiplicative ideal theory to develop explicit formulas for the core of ideals in various classes of integral domains (not necessarily Noetherian). We also investigate the existence of minimal reductions (originally established by Rees and Sally for local Noetherian rings). All results are illustrated by original examples in Noetherian and non-Noetherian settings, where we explicitly compute the core and validate some open questions recently raised in the literature.

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

Abstract. In this paper, we introduce and study the w-weak globaldimension w-w.gl.dim(R) of a commutative ring R. As an application, itis shown that an integral domain R is a Pru¨fer v-multiplication domainif and only if w-w.gl.dim(R) 6 1. We also show that there is a largeclass of domains in which Hilbert’s syzygy Theorem for the w-weak globaldimension does not hold. Namely, we prove that if Ris an integral domain(but not a field) for which the polynomial ring R[x] is w-coherent, thenw-w.gl.dim(R[x]) = w-w.gl.dim(R). 1. IntroductionRecall that a Pru¨fer domain is an integral domain in which every finitelygenerated ideal is invertible. It is well-known that the concept of Pru¨fer do-mains has played a central role in the development of the classical ideal theory.An important generalization of the Pru¨fer domain notion is that of a Pru¨ferv-multiplication domain (PVMD). This notion comes from multiplicative idealtheory, and various ideal-theoretic properties of it have been considered bymany authors, see for example [1, 2, 4, 7, 9, 10, 13, 14, 15]. From the homolog-ical algebra point of view, Pru¨fer domains are exactly the integral domains ofweak global dimension at most one. The original motivation for this work is toprovide a homological algebra characterization of PVMDs. To do so, we needthe notion of a w-flat module. Recently, modules of this type have receivedattention in several papers in the literature, see for example [3, 11, 21, 22].Throughout, R denotes a commutative ring with an identity element and allmodules are unitary.Now, we review some definitions and notation. Let J be an ideal of R.Following [25], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J isfinitely generated and the natural homomorphism ϕ : R → J

On the geometry of Prüfer intersections of valuation rings

Let $F$ be a field, let $D$ be a subring of $F$ and let $Z$ be an irreducible subspace of the space of all valuation rings between $D$ and $F$ that have quotient field $F$. Then $Z$ is a locally ringed space whose ring of global sections is $A = \bigcap_{V \in Z}V$. All rings between $D$ and $F$ that are integrally closed in $F$ arise in such a way. Motivated by applications in areas such as multiplicative ideal theory and real algebraic geometry, a number of authors have formulated criteria for when $A$ is a Pr\"ufer domain. We give geometric criteria for when $A$ is a Pr\"ufer domain that reduce this issue to questions of prime avoidance. These criteria, which unify and extend a variety of different results in the literature, are framed in terms of morphisms of $Z$ into the projective line ${\mathbb{P}}^1_D$

Star operations on modules

We present a theory of (semi)star operations for torsion-free modules. This extends the analogous theory of star operations on domains as in [R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972] and its generalization to semistar operations studied in [A. Okabe, R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994) 1–21], and recovers some closure operations defined on modules. We investigate some properties of (semi)star operations on a given module over a domain D and their relation with the properties of some classes of semistar operations induced on D. Among other things, this leads to a connection between semistar operations on the D-module and localizing systems on the domain D.

Module-Theoretic Characterizations of t-Linkative Domains

If R is an integral domain, we extend to any R-module the notion of semi-divisorial envelope, or w-envelope, defined by Wang and McCasland for torsion-free R-modules, and introduce and study the related notions of co-semi-divisoriality and w-nullity. These concepts are then used to give new module-theoretic characterizations of t-linkative domains, a class of domains widely considered in multiplicative ideal theory.

Nagata Rings, Kronecker Function Rings, and Related Semistar Operations

In 1994, Matsuda and Okabe introduced the notion of semistar operation. This concept extends the classical concept of star operation (cf. for instance, Gilmer's book (Gilmer, R. (1972). Multiplicative Ideal Theory. New York: Marcel Dekker) and, hence, the related classical theory of ideal systems based on the works by W. Krull, E. Noether, H. Prüfer and P. Lorenzen from 1930's. Fontana and Loper investigated properties of the Kronecker function rings which arise from arbitrary semistar operations on an integral domain D (Fontana M., Loper K. A. (2001a). Kronecker function rings: a general approach. In Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes Pure Appl. Math. 220, Marcel Dekker, pp. 189–205 and Fontana, M., Loper, K. A. (2001b). A Krull-type theorem for the semistar integral closure of an integral domain. ASJE Theme Issue “Commutative Algebra” 26:89–95). In this paper we extend that study and also generalize Kang's notion of a star Nagata ring to the semistar setting (Kang, B. G. (1987). ⋆-Operations on Integral Domains. Ph.D. dissertation, Univ. Iowa and Kang, B. G. (1989). Prüfer v-multiplication domains and the ring R[X] N v , J. Algebra 123: 151–171). Our principal focuses are the similarities between the ideal structure of the Nagata and Kronecker semistar rings and between the natural semistar operations that these two types of function rings give rise to on D.

Ideal Divisibility Monoids

Inspired by the monograph of Larsen/McCarthy, [26], in [10] and [11] the author started a series of articles concerning abstract multiplicative ideal theory along the problem lines of [26]. In this paper we turn to multiplicative lattices having the left Priifer property, that is to m-lattices satisfying the implication a1 + … + an ⊇ B ⇒ a1 +… + an ¦ℓ B or even the multiplication property A ⊇ B ⇒ A ¦B, respectively. Clearly, studying such structures includes studying substructures of d-semigroups.

Archimedan Prüfer Structures Alias Filter Archimedean d-Semigroups

Inspired by the monograph of {Larsen/McCarthy} the author\break started a series of articles concerning abstract multiplicative ideal theory along the lines of [23]. In the present paper we turn to archimedean Pr\"ufer structures, that is to algebraic \(m\)-lattices satisfying the two implications \(\left( P \right)a_1 + ... + a_n \supseteqq B \Rightarrow a_1 + ... + a_n |B\) and \(\left( A \right)A^n \supseteqq B\left( {\forall n \in N} \right) \Rightarrow AB = B = BA\) . Since these properties imply commutativity we start from commutative structures without real loss of generality and give characterizations of arbitrary and also of special Archimedean Pr\"ufer structures.