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.     

Integral domains having a unique Kronecker function ring

We study the class of integrally closed domains having a unique Kronecker function ring, or equivalently, domains in which the completion (or b-operation) is the only e.a.b star operation of finite type. Such domains are a generalization of Prüfer domains and have fairly simple sets of valuation overrings. We give characterizations by studying valuation overrings and integral closure of finitely generated ideals. We provide new examples of such domains and show that for several well-known classes of integral domains the property of having a unique Kronecker function ring makes them fall into the class of Prüfer domains.     

Semistar Linkedness and Flatness,Prüfer Semistar Multiplication Domains

Abstract

In 1994, Matsuda and Okabe introduced the notion of semistar operation, extending the “classical” concept of star operation. In this paper, we introduce and study the notions of semistar linkedness and semistar flatness which are natural generalizations, to the semistar setting, of their corresponding “classical” concepts. As an application, among other results, we obtain a semistar version of Davis' and Richman's overring-theoretical theorems of characterization of Prüfer domains for Prüfer semistar multiplication domains.     

Semistar ascending chain conditions over power series rings

Ricerche di Matematica - Let D be an integral domain. We associate to a semistar operation $$\star $$ on D, a semistar operation $$*$$ on D[[X]]. We prove that if D satisfies the $$\star _f$$...     

Star stable domains

We introduce and study the notion of ?-stability with respect to a semistar operation ? defined on a domain R; in particular we consider the case where ? is the w-operation. This notion allows us to generalize and improve several properties of stable domains and totally divisorial domains.     

Universally Catenarian Integral Domains,Strong S-Domains and Semistar Operations

Let D be an integral domain and a semistar operation stable and of finite type on it. In this article, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over D. We introduce and investigate the notions of -universally catenarian and -stably strong S-domains and prove that, every -locally finite dimensional Prüfer -multiplication domain is -universally catenarian, and this implies -stably strong S-domain. We also give new characterizations of -quasi-Prüfer domains introduced recently by Chang and Fontana, in terms of these notions.     

Morita duality and graded rings

ABSTRACT

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G ₁₂ , G ₂₄ , G ₂₅ and G ₂₆ . Then we give representatives of all the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups.     

Towards a classification of stable semistar operations on a Prüfer domain

We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain R has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of R.     

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗-unique representation domain (∗-URD) if each ∗-invertible ∗-ideal of R can be uniquely expressed as a ∗-product of pairwise ∗-comaximal ideals with prime radical. When ∗ is the t-operation we call the ∗-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19-29] and Brewer-Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999-6010], we give conditions for a ∗-ideal to be a unique ∗-product of pairwise ∗-comaximal ideals with prime radical and characterize ∗-URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.     

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lovász and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N₀-, N-, or N₊-rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N₀- or the N-rank. Hence we obtain the ranks for these graphs, and we also present some graph-minor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N₊-ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N₊-rank of a graph.Research of these authors was supported in part by a PREA from Ontario, Canada and research grants from NSERC.Mathematics Subject Classification (2000): 0C10, 90C22, 90C27, 47D20     

Vector subspaces of finite fields and star operations on pseudo-valuation domains

Given an extension of finite fields $F\subseteq L$, we study the number of the equivalence classes of F-vector subspaces of L modulo multiplication by elements of L, obtaining an exact formula and some bounds. We then apply the results obtained to the study of the set of F-star operations on L, which correspond to star operations on a pseudo-valuation domain R.     

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.     

Uppers to Zero in Polynomial Rings and Prüfer-Like Domains

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e., its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content c _D (g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with c _D (g) ^v  = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this article, given a semistar operation ☆ in the sense of Okabe–Matsuda, we introduce the ☆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

Prüfer ⋆–multiplication domains and ⋆–coherence

Abstract The purpose of this paper is to deepen the study of the Prüfer ⋆–mul-tiplication domains, where ⋆ is a semistar operation. For this reason, we introduce the ⋆–domains, as a natural extension of the v-domains. We investigate their close relation with the Prüfer ⋆-multiplication domains. In particular, we obtain a characterization of Prüfer ⋆-multiplication domains in terms of ⋆–domains satisfying a variety of coherent-like conditions. We extend to the semistar setting the notion of -domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the –domains, the Prüfer ⋆-multiplication domains coincide with the ⋆-domains. Keywords: Star and semistar operation, Prüfer (⋆-multiplication) domain, -domain, Localizing system, Coherent domain, Divisorial and invertible ideal Mathematics Subject Classification (2000): 13F05, 13G05, 13E99     

Acyclic and star colorings of cographs

An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus 1, and the pathwidth plus 1 are all equal for cographs.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.