Download Abstracts of Papers

ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
OTHMAN ECHI
My research papers are essentially in the following areas: Commutative
Algebra, Elementary Number Theory, Foliation Theory, General Topology,
Theory of categories(Categorical Algebra and Categorical Topology).
Year 1993
1. Othman, Echi. C-anneaux, E-anneaux et formule de la dimension. [Crings, E-rings and dimension formula] Portugaliae Matematica 50 (1993),
no. 3, 277–295.
In this paper, we introduce, in a general, not necessarily Noetherian context, a weaker version of the catenarity condition (the n-catenarity condition)
and some weaker versions of the universal catenarity property (C-property
and E-property). The relations among the above conditions are clarified, and
examples are given in order to show that the new classes of domains satisfying
these properties are distinct.
Year 1994
2. Echi, Othman. Transfert de la notion de C-anneaux aux produits fibrs.
[The notion of C-ring applied to pullbacks] Portugaliae Matematica 51
(1994), no. 1, 13–23.
Let R be a finite-dimensional integral domain and let X1 , X2 , · · · , Xn be
indeterminates over R. We call R a C-domain if, for each integer n ≥ 1
and for each pair P ⊂ Q of adjacent primes of R[X1 , X2 , · · · , Xn ] one has
ht(Q/p[X1 , X2 , · · · , Xn ]) = ht(P/p[X1 , X2 , · · · , Xn ]) + 1, where p = P ∩ R.
An integral domain R is a C-domain if and only if R is a totally Jaffard
domain and for each pair P ⊂ Q of adjacent primes of R[X1 , X2 , · · · , Xn ] one
has 1 − (ht(Q) − ht(P )) = ht(q/p) − (ht(q) − ht(p)), where p = P ∩ R and
q = Q ∩ R. Therefore, a catenarian C-domain is the same as a universally
1
2
OTHMAN ECHI
catenarian domain. The main result of this paper is the following: Let T be
an integral domain, M a maximal ideal of T , K = R/M the residue field of T
in M , φ : R → K the canonical surjective homomorphism, D a subring of K
and R = φ−1 (D). If T and D are C-domains and tr. deg .D K = 0, then R is a
C-domain.
3. Ayache, Ahmed; Cahen, Paul-Jean; Echi, Othman. Intersection de produits
fibrés et formule de la dimension. Communications in Algebra 22 (1994),
no. 9, 3495–3509.
Let T0 be a domain with maximal ideal M0 , k0 = T0 /M0 , φ : T0 → k0
the canonical surjection, and R0 = φ−1 (D0 ), where D0 is any subring of k0 .
The pullback R0 is said to be the ring of the construction (T0 , M0 , D0 ). Let
M1 , · · · , Mn be finitely many maximal ideals of a domain T . For each i, 1 ≤
i ≤ n, let Di be a subring of ki = T /Mi and let R be the intersection of rings
of constructions (T, Mi , Di ). The main purpose of this paper is to generalize
some classical results of transfer properties –established on pullbacks of the
type R0 – to R. Recall that, usually, intersections of pullbacks are taken into
consideration to construct examples and counterexamples.
4. Echi, Othman. The altitude formula. Commutative ring theory (Fs, 1992),
99–103, Lecture Notes in Pure and Applied Mathematics, 153, Dekker,
New York, 1994.
Let A be a commutative integral domain. One says that A satisfies the
altitude formula provided that, for every domain B containing A and finitely
generated as an A-algebra, and for every prime ideal P of B, one has ht(P ) =
ht(p)+tr degA (B)−tr degA/p (B/P ), where p = P ∩A. The following conditions
are shown to be equivalent: (1) A/I satisfies the altitude formula for every
prime ideal I of A. (2) For every n ≥ 1 and every pair P ⊂ Q of prime ideals
in the polynomial ring A[X1 , · · · , Xn ] one has
ht(Q/P ) − [ht(Q) − ht(P )] = ht(q/p) − [ht(q) − ht(p)],
where p = P ∩ A and q = Q ∩ A. Also, A satisfies the altitude formula if and
only if the identity in (2) holds whenever P ⊂ Q and P ∩ A = (0).
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
3
Year 1995
5. Ayache, Ahmed; Cahen, Paul-Jean; Echi, Othman. Valuative heights and
infinite Nagata rings. Communications in Algebra 23 (1995), no. 5, 1913–
1926.
Let R be a commutative ring with multiplicative identity. We prove that
if R has locally finite valuative dimension, then the Nagata ring R(∞) with
infinite many indeterminates is a stably strong S-ring. We also characterize
Jaffard properties in terms of Nagata rings. A link between the catenarity and
the v-catenarity of R(∞) is established.
6. Echi, Othman Sur les hauteurs valuatives. Bollettino della Union Matematica Italiana B (7) 9 (1995), no. 2, 281–297.
For R a commutative ring with identity, dimv R denotes its valuative dimension (i.e. the limit of the sequence (dim R[n] − n)). For a prime ideal p of R,
the valuative height of p, denoted by htv p, is the valuative dimension of the
localization Rp (i.e. the limit of the sequence (ht p[n])). We introduce a ”valuative version” of the so-called altitude formula; that is, R satisfies the valuative
altitude formula if for each finitely generated R-algebra T and for each prime
ideal q of T there holds htv q = htv (q ∩ R) + t.d.(T : R) − t.d.(T /q : R/(q ∩ R)),
where t.d. denotes transcendence degree. Most of the paper (Sections 2, 3 and
4) is devoted to various properties of this concept, whereas, in the first section,
a ”valuative version” of Jaffard’s theorem of the special chain is stated.
Year 1996
7. Ayache, Ahmed; Cahen, Paul-Jean; Echi, Othman. Anneaux quasi-prfriens
et P -anneaux. Bollettino della Union Matematica Italiana B (7) 10
(1996), no. 1, 1–24.
A quasi-Prfer ring A is defined by the following property: if Q is a prime
ideal in a polynomial ring A[X1 , · · · , Xn ], p a prime ideal of A and Q ⊂
p[X1 , · · · , Xn ] then Q = (Q ∩ A)[X1 , · · · , Xn ]. An integral domain is quasiPrfer if and only if its integral closure is Prfer. Furthermore, each quasi-Prfer
domain is a stably strong S-domain. Let Q be a prime ideal in a polynomial
4
OTHMAN ECHI
ring A[X1 , · · · , Xn ] and let q = Q ∩ A; the relative height of Q, denoted here
by ²Q , is the maximal length of the chains of prime ideals of A[X1 , · · · , Xn ]
between q[X1 , · · · , Xn ] and Q. The main result of the present paper is a characterization of quasi-Prfer rings by means of the notion of relative height: A
is quasi-Prfer if and only if for each pair Q ⊂ P of prime ideals in a polynomial ring A[X1 , · · · , Xn ], such that P is minimal among all the prime ideals
of A[X1 , · · · , Xn ] containing Q + (P ∩ A)[X1 , · · · , Xn ], then ²Q = ²P . In order
to prove this result, we give a new and neat proof of Nagata’s theorem concerning a general inequality relation between relative heights. A section of the
present paper is devoted to investigating the relation between quasi-Prfer rings
and various conditions of catenarity. Finally, quasi-Prfer domains (like Prfer
domains) can be characterized by means of their overrings; in particular, we
prove that the following properties are equivalent for an integral domain: (i)
quasi-Prfer, (ii) each overring is a stably strong S-domain, (iii) each overring
satisfies the dimension inequality.
8. Bouacida, Ezzeddine; Echi, Othman; Salhi, Ezzeddine. Topologies associes
une relation binaire et relation binaire spectrale. Bollettino della Union
Matematica Italiana B (7) 10 (1996), no. 2, 417–439.
We define and study the notion of spectral binary relation, which generalizes
the notion of spectral ordered set (i.e., a set isomorphic to the spectrum of a
ring), and we generalize some previous results due to Bouvier and Fontana.
Using binary relations, we characterize the topological spaces in which the
intersection of some family of open sets is an open set (Alexandroff spaces).
Year 1997
9. Belad, Karim; Cherif, Bahri; Echi, Othman. On spectral binary relation.
Commutative ring theory (Fs, 1995), 79–88, Lecture Notes in Pure and
Applied Mathematics, 185, Dekker, New York, 1997.
We present, here, some ideas related to that of M. Hochster[Trans. Amer.
Math. Soc. 142 (1969), 43–60], W. J. Lewis and J. Ohm [Canad. J. Math.
28 (1976), no. 4, 820–835] and E. Bouacida, Echi and E. Salhi [Boll. Un.
Mat. Ital. B (7) 10 (1996), no. 2, 417–439]. A relation R on X induces
an equivalence relation T (R), a map ρ : X → X/T (R), and an order ≤R on
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
5
X/T (R). A topology τ on X is quasi-spectral when it satisfies all of Hochster’s
spectral conditions except for requiring T0 . We show that (X, R) is spectral
if and only if (X/T (R), ≤R ) is a spectral ordered space. (There is an order
compatible quasi-spectral topology on X.) We extend this result to disjoint
unions. If Rl (X) = ρ−1 ( ] ←, x]), then (xi , i ∈ I) is an R-chain when {Rl (xi )}
is a chain under inclusion. For spectral relations, R-chains must have an Rsupremum and an R-infimum and each pair of R-chain elements must contain
adjacent elements.
R is a spectral relation on X iff its inverse relation is also spectral.
For components of finite (Krull) dimension which satisfy certain finiteness
conditions on the disjoint union, that union of relations is spectral.
10. Bouacida, Ezzeddine; Echi, Othman; Salhi, Ezzeddine. Nonfinite heights.
Commutative ring theory (Fs, 1995), 113–130, Lecture Notes in Pure and
Applied Mathematics, 185, Dekker, New York, 1997.
All rings considered are commutative. If α is an ordinal number, we use
transfinite induction to define what it means for a prime ideal p of a ring R
to be of height α; if each p ∈ Spec(R) has a height, ht(p), in the above sense,
then dim(R), the Krull dimension of R, is defined as sup({ht(p)}). A typical
result, Theorem 2.1, concerns the height of a prime ideal in a polynomial ring
in finitely many variables, and is the analogue of a fundamental result in the
classical case. In comparison with the classical case of heights being either
finite or ”∞”, the above definition presents some anomalous behavior. For
instance : ht(p) need not be the supremum of lengths of chains of prime ideals
with maximal element p; a valuation ring need not have a Krull dimension;
and the class of rings having Krull dimension is not stable under formation
of unions of increasing sequences. Other results include the fact that for each
ordinal α, there exists a ring with Krull dimension α. The paper also includes
consideration of heights and coheights in arbitrary (partially) ordered sets and
a study of dimV (R), the valuative dimension of a ring in terms of valuative
heights of prime ideals (which are also defined as suitable ordinals). It is shown
that if dim(R) = α and α is a limit ordinal, then dimV (R) = α; the assertion
is false in general for a nonlimit ordinal α.
6
OTHMAN ECHI
Year 1999
11. Bouacida, Ezzeddine; Echi, Othman; Salhi, Ezzeddine. Foliations, spectral
topology, and special morphisms. Advances in commutative ring theory (Fez,
1997), 111–132, Lecture Notes in Pure and Applied Mathematics, 205,
Dekker, New York, 1999.
Let F be a codimension-one foliation of class C r (r ≥ 0), transversely oriented on a closed connected manifold M . Set X = M/F, the space of classes
of leaves of F(the class of a leaf F is the union of the leaves G such that
F = G). Suppose that F has a well-defined height. Let X0 be the union of
open subsets of X homeomorphic to R or S 1 . In [J. Math. Soc. Japan 52
(2000), no. 2, 447–464], we have shown that X \ X0 is a spectral space. Let
Z = M/F be the space of leaves of F and Z0 be the union of open subsets of
Z homeomorphic to R or S 1 . In this paper we introduce the notion of special
morphism used to prove that Z \ Z0 is a quasi-spectral space.
Year 2000
12. Bouacida, Ezzeddine; Echi, Othman; Salhi, Ezzeddine. Feuilletages et
topologie spectrale. Journal of Mathematical Society of Japan 52 (2000),
no. 2, 447–464.
Let F be a codimension-one foliation, transversally oriented, of class C r (r ≥
0) on a connected closed manifold M . The class of a leaf F of F is defined to
be the union of all leaves G with F = G. Let X be the space of classes of leaves
in M and let X0 be the union of open subsets of X which are homeomorphic
to R or to S 1 . In this paper we prove that if the level of F is well-defined (in
the sense of E. Salhi[C. R. Acad. Sci. Paris Sr. I Math. 301 (1985), no. 5,
219–222]), then X \ X0 is a spectral space.
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
7
13. Nasr, Mabrouk Ben; Echi, Othman; Izelgue, Lahoucine; Jarboui, Noman.
Pairs of domains where all intermediate domains are Jaffard. Journal of
Pure Applied Algebra 145 (2000), no. 1, 1–18.
In this paper we deal with the study of pairs of rings where all intermediate
rings are Jaffard. Furthermore, we introduce a new invariant allowing us to
compute the number of Jaffard domains between any given extension of integral
domains A ⊆ B. We also give a new characterization of valuation domains
and one-dimensional Prfer domains and provide many examples to illustrate
the theory.
14. Ameziane, Souad; Echi, Othman; Yengui, Ihsen. Direct systems of localizations of polynomial rings. Acta Scientarum Mathematicarum (Szeged)
66 (2000), no. 3-4, 465–476.
All rings are commutative and with a unit element. Let
R[n] = R[x1 , x2 , . . . , xn ], R(n) = R(x1 , x2 , . . . , xn ),
and Rhni = Rhx1 , x2 , . . . , xn i be, respectively, the polynomial ring, the Nagata
ring and the Serre conjecture ring. It is known that Rh∞i and R(∞) are
stably strong S-domains if they are locally finite-dimensional. In this paper,
we prove analogous results for certain direct systems with lifting property. It
is shown that if R is an integral domain and (Sj−1 R[Λj ], fkj ) is a direct system
of rings indexed by a direct set (I, ≤), where fkj is an R-homomorphism,
Λj a set of indeterminates over R, and Sj a multiplicative subset of R[Λj ],
and if A = lim Sj−1 R[Λj ] is locally finite-dimensional and the transcendental
−→
degree of [A : R] = ∞, then A is a stably strong S-domain. We also give
another characterization of rings satisfying the valuative altitude formula. Two
examples of increasing sequences of rings are given.
15. Belaid, Karim; Cherif, Bahri; Echi, Othman. Quasi-spectral binary relations and ordered disjoint unions. Journal of Mathematical Sciences
(Calcutta) 11 (2000), no. 2, 139–157.
Let f be a mapping: X → X. As a binary relation, its transitive closure T
induces a partial ordering ≤ on the quotient set X/T . The main result of this
paper is that (X/T, ≤) is isomorphic to the spectrum of some ring if and only
if each of its totally ordered subsets is finite.
8
OTHMAN ECHI
16. Echi, Othman. A topological characterization of the Goldman prime spectrum of a commutative ring. Communications in Algebra 28 (2000), no.
5, 2329–2337.
A prime ideal p of a commutative ring R is said to be a Goldman ideal
(or a G-ideal) if there exists a maximal ideal M of the polynomial ring R[X]
such that p = M ∩ R. A topological space is said to be Gold-spectral if it is
homeomorphic to the space Gold(R) of G-ideals of R(Gold(R) is considered a
subspace of the prime spectrum Spec(R) equipped with the Zariski topology).
We give here a topological characterization of Gold-spectral spaces.
Year 2001
17. Ayache, Ahmed; Ben Nasr, Mabrouk; Echi, Othman; Jarboui, Nomen.
Universally catenarian and going-down pairs of rings. Mathematische Zeitschtrif
238 (2001), no. 4, 695–731.
Let P be a ring-theoretic property and R ⊆ S be an extension of commutative rings. We say that (R, S) is a P-pair if each intermediate ring T ,
with R ⊆ T ⊆ S, satisfies P. In case of a pair (R, L), where L is a field, we
obtain very satisfactory necessary and sufficient conditions for a pair of this
kind to be a P-pair, when P is one of the following properties: (stably) strong
S(eidenberg)-domain, universally catenarian, Jaffard, locally (and totally) Jaffard. For instance, (R, L) is a Jaffard pair if and only if either R is a field and
tr.d.R (L) ≤ 1 or R is a quasi-Prfer domain (i.e. the integral closure R0 of R is
a Prfer domain) and tr.d.R (L) = 0.
Particular interest is devoted to the going-down pairs. A pair (R, S) is called
a GD pair if every extension A ⊆ B has the property GD, when R ⊆ A ⊆ B ⊆
S. Among the several results proved in this context, we mention the following:
Let R ⊆ S be a ring extension. Then R[X] ⊆ S[X] is a GD pair if and only if
R ⊆ S is an integral extension, the canonical morphism Spec(S) → Spec(R)
is a bijection and qf(S/Q) is a purely inseparable extension of qf(R/(Q ∩ R)),
for each prime ideal Q of S. Moreover, in general, if (R, S) is a GD pair, then
dim(S) ≤ dim(R) + 1; in particular if (R, qf(R)) is a GD pair then dimv (R) ≤
dim(R) + 1.
In case R ⊆ S is an algebraic ring extension and R is a going-down domain,
then the following properties are equivalent: (i) (R, S) is a universally catenarian pair; (ii) (R, S) is a (stably) strong S-pair; (iii) (R, S) is a locally Jaffard
pair; (iv) R is a locally Jaffard domain.
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
9
In a final section, we investigate various examples in order to show the
essentiality of the hypotheses in several of the results proved in the paper.
Year 2003
18. Echi, Othman; Jarboui, Nomen. On residually integrally closed domains.
Demonstratio Mathematica 36 (2003), no. 3, 543–550.
A domain R is called residually integrally closed if R/p is an integrally closed
domain for each prime ideal p of R. We show that residually integrally closed
domains satisfy some chain conditions on prime ideals. We give a characterization of such domains in case they contain a field of characteristic 0. Section 3
deals with domains R such that R/p is a unique factorization domain for each
prime ideal p of R; these domains are shown to be PID. We also prove that
domains R such that R/p is a regular domain are exactly Dedekind domains.
19. Bouacida, Ezzeddine; Echi, Othman; Picavet, Gabriel; Salhi, Ezzeddine.
An extension theorem for sober spaces and the Goldman topology. International Journal of Mathematics and Mathematical Sciences 2003, no.
51, 3217–3239.
Goldman points of a topological space are defined in order to extend the
notion of prime G-ideals of a ring. We associate to any topological space a
new topology called Goldman topology. For sober spaces, we prove an extension theorem of continuous maps. As an application, we give a topological
characterization of the Jacobson subspace of the spectrum of a commutative
ring. Many examples are provided to illustrate the theory.
20. Echi, Othman. Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces. Bollettino della Union Matematica Italiana B Artic. Ric.
Mat. (8) 6 (2003), no. 2, 489–507.
Recall from [A. Grotendik and Ž. A. Dieudonné, Uspehi Mat. Nauk 27
(1972), no. 2(164), 135–148] (a) continuous map f : X → Y is called a quasihomeomorphism if, for each open subset U ⊆ X, there exists a unique open
subset V ⊆ Y such that U = f −1 (V ); (b) a topological space X is called
a sober space if each nonempty irreducible closed subset of X has a unique
10
OTHMAN ECHI
generic point; (c) the set sX of all irreducible closed subsets of X can be
naturally endowed with a topology (induced by the topology of X); sX is
called the sobrification of X and the continuous map q : X → sX, x 7→ {x},
is a quasi-homeomorphism. If R is a ring, Gold(R) denotes the set of all
the G(oldman)-primes of R (i.e. the prime ideals P of R such that R/P
is a G(oldman)-domain [cf. I. Kaplansky, Commutative rings, Allyn-Bacon,
Boston, Mass., 1970]). An ordered set (X, ≤) is said to be a Goldspectral set
if there exists a commutative ring R such that (X, ≤) is order isomorphic to
(Gold(R), ⊆). One of the results proved in this paper is the following: Let
X be a T0 -space, let Gold(X) be the set of all locally closed points of X,
and let q : X → sX be the canonical embedding of X in its sobrification, then
q(Gold(X)) = Gold(sX). Application of this result to the Goldspectral sets is
discussed.
The Jacobson prime spectrum of a commutative ring R is the set Jac(R) of
all prime ideals of R that are intersections of maximal ideals of R. A topological
space X is called a Jacobson space if the set of its closed points X0 is strongly
dense in X. If X is a topological space, we denote by Jac(X) the largest
subspace of X in which X0 is strongly dense, clearly X is a Jacobson space if
and only if X = Jac(X). For a T0 -space X, having a basis of quasi-compact
open subsets, X is a Jacobson space if and only if X0 = Gold(X).
If X = Spec(R) is the Zariski prime spectrum of some commutative ring R,
then Jac(Spec(R)) coincides with Jac(R) and Gold(Spec(R)) coincides with
Gold(R). A Jacspectral space is a topological space homeomorphic to Jac(R)
for some commutative ring R. One of the main theorems proved in the present
paper shows that the disjoint union of a family of ordered sets is a Jacspectral
set if and only if each ordered set of the family is Jacspectral.
21. Echi, Othman. Topological characterizations of some subspaces of a spectral space. Questions and Answers in General Topology 21 (2003), no.
2, 109–123.
Let Spec(R) be the prime spectrum of a commutative ring R with identity,
endowed with the Zariski topology. M. Hochster [Trans. Amer. Math. Soc.
142 (1969), 43–60] characterized topological spaces which are homeomorphic
to the prime spectrum Spec(R) of some ring R. We obtain a more general
theorem which implies his previous result cited above. He also gives an intrinsic
topological characterization of the topological spaces which are homeomorphic
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
11
to the Jacobson subspace of the prime spectrum Spec(R) of some ring R. A
characterization of the sober spaces by means of an extension-type property is
obtained as well. Many questions are, also, stated.
Year 2004
22. Belaid, Karim; Echi, Othman; Gargouri, Riyadh. A-spectral spaces.
Topology and its Applications 138 (2004), no. 1-3, 315–322.
A topological space X is said to be spectral if and only if it satisfies the
following four conditions: (1) It is sober; (2) it is quasi-compact; (3) the quasicompact open subsets form a basis for the topology of X; (4) the family of
quasi-compact open subsets of X is closed under finite intersections. The
given conditions describe those topological spaces that are homeomorphic to
the prime spectrum of a commutative ring with identity endowed with the
Zariski topology. We characterize the topological spaces X whose Alexandroff
e is spectral. These spaces are called
extension or one-point-compactification X
A-spectral. (Here, as usual, for a topological space X, the one-point-extension
e = X ∪ {∞} is equipped with the topology whose members are the open
X
e such that X
e \ U is a closed quasi-compact
subsets of X and all subsets U of X
subset of X.)
23. Belaid, Karim; Echi, Othman. On a conjecture about spectral sets.
Topology and its Applications 139 (2004), no. 1-3, 1–15.
In the paper, [Canad. J. Math. 28 (1976), no. 4, 820–835], Lewis and Ohm
proved that the ordered disjoint union of arbitrarily many spectral sets is itself
a spectral set and they raised the question of whether the converse is valid. The
paper under review contributes to the study of this question. It is noted that
the question is equivalent to asking whether each D-component of a spectral
set is necessarily spectral. (Recall that if (X, ≤) is a poset and x ∈ X, then
the D-component of x is defined to be the intersection of the subsets of X that
are closed under both ≤ and ≥.) We obtain results for some new topologies
on D-components. In addition, they define a poset X to be an up-spectral
set (resp., a down-spectral set) if the result of adjoining to X a new unique
greatest (resp., smallest) element produces a spectral set. A topological space
X is called an up-spectral space (resp., down-spectral space) if X satisfies the
12
OTHMAN ECHI
axioms of a spectral space (as in [M. Hochster, Trans. Amer. Math. Soc. 142
(1969), 43–60)]) with the exception that X is not necessarily quasi-compact
(resp., that X, if irreducible, does not necessarily have a generic point). It
is proved that a poset (X, ≤) is an up-spectral set (resp., a down-spectral
set) if and only if there exists an up-spectral topology (resp., a down-spectral
topology) on X that is order-compatible with ≤. Thus, each spectral set is
both an up-spectral set and a down-spectral set. Among the many relevant
results in the paper, we state the following two. Any disjoint union of upspectral sets (resp., down-spectral sets) is itself an up-spectral set (resp., a
down-spectral set). A disjoint union of a family {Xλ } of topological spaces is
an up-spectral space if and only if each Xλ is an up-spectral space.
24. Echi, Othman; Gargouri, Riyadh. An up-spectral space need not be Aspectral. New York Journal of Mathematics 10 (2004), 271–277.
An A-spectral space is a space such that its one-point compactification is
a spectral space. An up-spectral space is defined to be a topological space
satisfying the axioms of a spectral space with the exception that X is not
necessarily compact. This paper deals with interactions between up-spectral
spaces and A-spectral spaces. An example of up-spectral space which is not
A-spectral is constructed.
25. Belaid, Karim; Echi, Othman; Lazaar, Sami. T(α,β) -spaces and the Wallman compactification. International Journal of Mathematics and Mathematical Sciences 2004, no. 65-68, 3717–3735.
Some new separation axioms are introduced and studied. We also deal
with maps having an extension to a homeomorphism between the Wallman
compactifcations of their domains and ranges
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
13
Year 2005
26. Echi, Othman; Itō, Munehiko. Up-spectral spaces and A-spectral spaces.
Questions and Answers in General Topology 23 (2005), no. 1, 15–26.
This paper is connected to the papers [E. Khalimsky, R. D. Kopperman
and P. R. Meyer, Topology Appl. 36 (1990), no. 1, 1–17]; [T. Y. Kong, R.
D. Kopperman and P. R. Meyer, Amer. Math. Monthly 98 (1991), no. 10,
901–917], which were concerned with applications of topology to theoretical
computer science.
The following results are proved:
– A topological space is up-spectral iff it is a sober space and has a basis of
quasi-compact open subsets which is closed under finite intersections.
– Each A-spectral space is up-spectral. 3. Let K be the space on the
integers Z such that a set U is open iff whenever x ∈ U is an even integer,
then x − 1, x + 1 ∈ U . Then K is an A-spectral space.
27. Belaid, Karim; Echi, Othman; Gargouri, Riyadh. Two classes of locally
compact sober spaces. International Journal of Mathematics and Mathematical Sciences 2005, no. 15, 2421–2427.
We deal with two classes of locally compact sober spaces, namely, the class
of locally spectral coherent spaces and the class of spaces in which every point
has a closed spectral neighborhood (CSN-spaces, for short). We prove that
locally spectral coherent spaces are precisely the coherent sober spaces with
a basis of compact open sets. We also prove that CSN-spaces are exactly the
locally spectral coherent spaces in which every compact open set has a compact
closure.
28. Ayache, Ahmed; Echi, Othman. The envelope of a subcategory in topology
and group theory. International Journal of Mathematics and Mathematical Sciences 2005, no. 21, 3387–3404.
Let C be a category and D a subcategory of C closed under isomorphisms.
The envelope of D in C is defined as the largest full subcategory of C in which
D is a reflective subcategory. In this paper a collection of results are presented
14
OTHMAN ECHI
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 re-proved. Finally,
attention is focused on the envelopes of one-object subcategories, and examples
are considered in the category of groups.
Year 2006
29. Ayache, Ahmed; Echi, Othman. Valuation and pseudovaluation subrings
of an integral domain. Communications in Algebra 34(2006), 2467-2483.
Let R, S be two rings. We say that R is a valuation subring of S (R is
a V D in S, for short) if R is a proper subring of S and whenever x ∈ S,
we have x ∈ R or x−1 ∈ R. We denote by N u(R) the set of all non unit
elements of a ring R. We say that R is a pseudo-valuation subring of S (R is
a P V in S, for short) if R is a proper subring of S and x−1 a ∈ R, for each
x ∈ S \ R, a ∈ N u(R). This paper deals with the study of valuation subrings
and pseudo-valuation subrings of a ring; interactions between the two notions
are also given. Let R be a P V in S; the Krull dimension of the polynomial
ring on n indetrminates over R is also computed.
30. Ayache, Ahmed; Dobbs, David E.; Echi, Othman. Reflection of some
quasi-local domains. Journal of Algebra and its Applications 5 (2006),
no. 2, 201–213.
If (R, M ) and (S, N ) are quasi-local domains and f : R −→ S is a ring
homomorphism, then f is said to be a strong local homomorphism if f (M ) =
N . Let PVD be the category whose objects are the pseudo-valuation domains
that are not fields and whose morphisms are the strong local homomorphisms;
let VD be the full subcategory of PVD whose objects are all the valuation
domains that are not fields. Then VD is shown to be a reflective subcategory
of PVD. This result is extended by obtaining a reflective conclusion for a
category whose class of objects properly contains all pseudo-valuation domains.
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
15
31. Ayache, Ahmed; Dobbs, David E; Echi, Othman. Universal Mapping
Properties of some Pseudo-valuation Domains and Related Quasi-local Domains. International Journal of Mathematics and Mathematical Sciences 2006, Article ID 72589, Pages 1-12.
If (R, M ) and (S, N ) are quasi-local (commutative integral) domains and
f : R −→ S is a (unital) ring homomorphism, then f is said to be a strong
local homomorphism (resp., radical local homomorphism) if f (M ) = N (resp.,
f (M ) ⊆ N and for each x ∈ N , there exists a positive integer t such that xt ∈
f (M )). It is known that if f : R −→ S is a strong local homomorphism where
R is a pseudo-valuation domain that is not a field and S is a valuation domain
that is not a field, then f factors via a unique strong local homomorphism
through the inclusion map iR from R to its canonically associated valuation
overring, (M : M ). Analogues of this result are obtained which delete the
conditions that R and S are not fields, thus obtaining new characterizations
of when iR is integral or radiciel. Further analogues are obtained in which
the “pseudo-valuation domain that is not a field” condition is replaced by the
APVDs of Badawi-Houston and the “strong local homomorphism” conditions
are replaced by “radical local homomorphism”.
32. Echi, Othman. On TD -spaces. Missouri Journal of Mathematical
Sciences 18(2006), no. 1, 54–60.
This paper deals with some new properties of TD -spaces. These properties are used in order to give an intrinsic topological characterization of the
Goldman spectrum of a commutative ring.
33. Echi, Othman. Binomial coefficients and Nasir al-Din al-Tusi. Scientific
Research and Essay Vol. 1 (2), pp. 028 -032, November 2006.
We give a historical note about the scientist Nasir al-Din al-Tusi legitimating
the introduction of a new concept related to binomial coefficients. Al-Tusi
binomial coefficients and binomial formulas are introduced and studied.
34. Echi, Othman. Networks of Morphisms. Missouri Journal of Mathematical Sciences 19(2007) n2, 93 - 105.
This paper deals with some classes of morphisms in a category which we
call networks of morphisms. These networks are linked with full reflective
subcategories. The notion of saturated subcategory is introduced and studied.
Each reflective subcategory is showed to be saturated.
16
OTHMAN ECHI
Year 2007
35. Echi, Othman; Marzougui, Habib; Salhi, Ezzeddine. Problems from the
Bizerte-Sfax-Tunis Seminar, in Open Problems in Topology II, E. Pearl
ed., Elsevier, Amsterdam, 2007, pp. 669-674.
The Bizerte–Sfax–Tunis seminar [BST Seminar “Algebra, Combinatorics,
Dynamical Systems and Topology”] is organized by two Tunisian research
groups “Algebra and Topology 03/UR/03-15” and “Dynamical Systems and
Combinatorics 99/UR/15-15”. Three meetings are held each academic year in
one of the Faculties of Sciences of Bizerte, Sfax or Tunis. The Seminar has
been founded, firstly, by Professor Ezzeddine Salhi since 1996 and has been
called Bizerte–Sfax Meeting. In March 2001, Othman Echi has got the position
of Professor at Faculty of Sciences of Tunis; and so the seminar is shared by
Bizerte, Sfax and Tunis. The goal of this seminar is to shed light on the latest
results obtained by the members of the two groups (in the areas of algebra,
algebraic topology, combinatorics, complex analysis, dynamical systems, foliation theory, topology). It is worth noting that an interesting link between
foliation theory and spectral topology has been discovered by three members of
the above two research groups (see Bouacida, Ezzeddine; Echi, Othman; Salhi,
Ezzeddine. Feuilletages et topologie spectrale. Journal of Mathematical
Society of Japan 52 (2000), no. 2, 447–464.).
This paper deals with eight problems in Topology which are proposed by
our seminar. These problems concern spectral spaces and some related topics;
the space of leaves of a foliation; dynamics of groups of homeomorphisms and
vector fields on surfaces.
36. Ayache, Ahmed; Dobbs, David E.; Echi, Othman. On Maximal non-accp
Subrings. Journal of Algebra and its Applications 6 (2007) n 5, 873-894.
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
ABSTRACTS OF RESEARCH ARTICLES OF OTHMAN ECHI
17
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.
Papers Accepted for Publication
37. Echi, Othman. Williams Numbers. Mathematical Reports of the
Academy of Sciences of the Royal Society of Canada (to appear).
Let N be a composite squarefree number; N is said to be a Carmichael
number if p − 1 divides N − 1 for each prime divisor p of N . In [On numbers
analogous to the Carmichael numbers, Canad. Math. Bull. 20(1977), 133 −
143], Williams (H.C) has stated an interesting problem of whether there exists
a Carmichael number N such that p + 1 divides N + 1 for each prime divisor
p of N . This is a long standing open question; and it is possible that there is
no such number.
For a given nonzero integer a, we call N an a-Korselt number if N is composite, squarefree and p − a divides N − a for all primes p dividing N . We will
say that N is an a-Williams number if N is both an a-Korselt number and a
(−a)-Korselt number.
Extending the problem of Willams, one may ask more generally, the following:
For a given nonzero integer a, is there an a-Williams number?
We give an affirmative answer to the above problem, for a = 3p, where p is
a prime number such that 3p − 2 and 3p + 2 are primes. We, also, prove that
each a-Williams number has at least three prime factors.
38. Adams, Mick; Belaid, Karim; Dridi, Lobna; Echi, Othman. Submaximal
Spaces and Spectral Spaces. Mathematical Proceedings of the Royal
Irish Academy (to appear).
A topological space X is said to be submaximal if every dense subset of X
is open. In this paper, descriptions of submaximal spectral spaces and Stone
submaximal spaces are given. Throughout, a number of illustrative examples
are given.
18
OTHMAN ECHI
39. Echi, Othman; Lazaar, Sami. Universal Spaces, Tychonoff and Spectral
Spaces. Mathematical Proceedings of the Royal Irish Academy (to
appear).
This paper deals with some universal spaces. The class of morphisms in
T OP orthogonal to all Tychonoff spaces is characterized. We also characterize
topological spaces X for which the universal Tychonoff space associated to X
is a spectral space.
40. Echi, Othman; Lazaar, Sami. Sober Spaces and Sober Sets. Missouri
Journal of Mathematical Sciences(to appear).
We introduce and study the notion of sober partially ordered sets. Some
questions about sober spaces are also stated.