Ahmed Ayache

Minimal Overrings of an Integrally Closed Domain

Abstract The main purpose of this paper is to characterize minimal overrings of an integrally closed domain R. We show that there exists a strong relationship between minimal overrings and the notion of ideal transforms. In particular, we prove that if T(M) = S(M) for each maximal ideal M, then there is a bijective correspondence between the set of invertible maximal ideals of R and the set of minimal overrings of R. This study enables us to produce several interesting applications concerning semi-local, Dedekind, Prüferian and Krull domains. Moreover, we investigate the spectrum of a minimal overring in comparison with the spectrum of R, and we determine whether the polynomial ring R[X 1, X 2,…, X n ] has a minimal overring.

ON MAXIMAL NON-ACCP SUBRINGS

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.

Radical valuatif et sous-extensions

Resume. Si D est un anneau integre, on considere Ies sous-extensions de D-algebres de type fini, notamment les anneaux compris entre D et l’anneau de polynomes D[n] en n indeterminees. Au prealable on montre qu’une extension de type fini R ⊆ T, ou T est noetheiien, verifie l’inegalite de la dimension. Puis on e;tudie les radicaux dimensionnels et valuatifs d’un anneau (respectivement intersection des ideaux maximaux de plus grande hauteur et des ideaux maximaux de plus grande hauteur valuative). On montre ensuite que, si D est un anneau de Jaffard, tous les anneaux compris entre D et D[n] sont de Jaffard. II en est de meme des sous-extensions de D-algebres de type fini lorsqu’en outre le radical dimensionnel de D est nul. On conclut avec les sous-extensions d’un corps k. Si tout ideal premier d’une sous-extension R se releve dans une k-algebre de type fini contenant R, on montre que R est localement et residuellement de Jaffard. Lorsque toute chaine de R se releve dans une telle extension, on montre queR ...

The envelope of a subcategory in topology and group theory

A collection of results are presented which are loosely centered around the notion of reflective subcategory. For example, it is shown that reflective subcategories are orthogonality classes, that the morphisms orthogonal to a reflective subcategory are precisely the morphisms inverted under the reflector, and that each subcategory has a largest “envelope” in the ambient category in which it is reflective. Moreover, known results concerning the envelopes of the category of sober spaces, spectral spaces, and jacspectral spaces, respectively, are summarized and reproved. Finally, attention is focused on the envelopes of one-object subcategories, and examples are considered in the category of groups.

About a conjecture on Nagata rings

Abstract This paper provides a counterexample to a conjecture on Nagata rings: a two-dimensional quasi-local domain such that, for every n , the Nagata ring A ( n ) is strong S (and also catenary) but for n ≥ 1, the polynomial ring A [ n ] is not strong S (nor catenary).

Valuative Heights and Infinite Nagata Rings

We prove here that if the ring R has locally finite valuative dimension then the Nagata ring with infinite indeterminates R(∞) is a stably strong S-ring. We also study the catenarity of R(∞)

Intersection de produits fibres et formule de la dimension

We generalize classical results of tranfer properties to finite intersections of pullbacks at various maximal ideals of a domain T; we also get new transfer results on the dimension formula.

Finite maximal chains of commutative rings

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.

Topological indices of the mk-graph

Abstract This paper is devoted to the study of a new graph, called -graph and denoted by , that arises from a simple graph G by k successive iterations of a special kind of construction. We provide some of its topological indices such as Wiener index, Hyper-Wiener index, Wiener polarity, Zagreb indices, Schultz and modified Schultz indices and Wiener-type invariant. Some Formulas concerning classical graphs are settled to show the scope of this study.

A Direct Proof of a Theorem concerning Treed Overrings

We improve and provide a direct proof of a theorem that answered a question by Dobbs about treed overrings. More precisely, we show that if is an integrally closed local domain and each of its proper overrings is treed, then is a valuation domain or is a PVD with associated valuation overring such that .