Faculty of Science - Mahidol University

Mahidol University Annual Research Abstracts, Vol. 33 301
idela of R to M can be extended to R. We now extend this notion to
modules. We call a module N an M-minijective module if every
homomorphism from a simple M- cyclic submodule of M to N can
be extended to M. In this note, we charactrize quasi-mininjective
modules and show that for a finitely generated quasi-mininjective
module M which is a Kasch module, there is a bijection between the
class of all maximal submodules of M and the class of all minimal
left ideals of its endomorphism ring S = End(M) if an donly if l S r M (K)
= K for any simple left ideal K of S. The results obtained by
Nihcolson and Yousif in mininjective rings are generalized
(East-west J. of Mathematics: Vol 5, No. 2 pp113-122.)
ON THE STRUCTURE OF SERIAL ARTINIAN
MODULES (NO. 793)
Nguyen Van Sanh
Department of Mathematics, Faculty of Science, Mahidol
University, E-mail : frnvs@mahidol.ac.th
Key words : artinian module, uniserial, serial indecomposable,
quasi-injective, quasi-projective
Let R be an associate ring with identity. A right R-module
M is called uniserial if the lattice of submodules is linear. M is called
serial if it is a direct sum of uniserial modules. The main result in
this note is the following Theorem : Let M be an Artinian right Rmodule.
Then the following conditions are equivalent :
(1) M is uniserial and S = End(M) is left Artinian;
(2) Every indecomposable module in σ [M] is quasi-injective;
(3) Every indecomposable module in σ[M] is quasi-projective;
(4) Every indecomposable quasi-projective module in σ[M] is
quasi injective;
(5) Every indecomposable quasi-injective module in σ[M] is
quasi projective
This theorem generalizes a result of Muller in 1969 which
saying that a right Artinian ring R is generalized uniserial if and
only if every indecomposable right R-module is quasi-injective.
RING OVER WHICH EVERY CYCLIC MODULE
HAS A S-CS INJECTIVE HULL (NO. 794)
Maliwan Tunapan, Sarapee Chairat, Chitlada Somsup and Dinh
Van Huynh
Department of Mathematics, Faculty of Science, Mahidol
University, E-mail : frnvs@mahidol.ac.th
Key words : cyclic module, e-CS injective hull, q2fd-ring, einjective,
CSI-ring, CSE-ring, semilocal, von Neumann regular
A ring R is called a right qfd-ring if R/I has finite Goldie
dimension for any right ideal I of R. In 2003, C. Faith studied a
class of rings such that the injective hull of every cyclic right Rmodule
is Σ-injective. He called such a ring a ring CSI-ring. In this
research work, we study the class of ring over which every cyclic
ringht R-module has a Σ -CS injective hull. We call them right CSErings.
The key Lemma in our note is that any right CSE-ring right
qfd, and therefore any right CSE-ring is right Goldie. By a result of
Camillo in 1975, we can see that any commutative CSE-ring is
Noetherian. The goal of our research is considerig in the case of
non-commutativity. Our main result is that a right CSE-ring R is
Noetherian under one of the following conditions:
(1) R is semilocal;
(2) R or R/rad (R) is right Kash;
(3) R/rad (R) is von Neumann regular
We also showed that a ring R is right Noetherian if the
injective hull of any countable generated semisimple right R-modules
in ?Σ -CS.
ON THE RELATIONS BETWEEN THE CLASSES
OF WEAKLY QF3-RINGS AND HEREDITARY
RINGS (NO. 795)
Maliwan Tunapan, Phan Dan and Nguyen Van Sanh
Department of Mathematics, Faculty of Science, Mahidol
University, E.-mail : frnvs@mahidol.ac.th
Key words : weakly QF3-ring, hereditary ring, injective, projective
Menabu Harada (1965, 1978, 1980) has studied some
generalizations of QF-rings by introducting two following conditions:
(*) Every non-small right R-module contains a non-zero
injective submodule;
(**) Every non-cosmall right R-module contains a non-zero
projective direct summand;
(***) Every indecomposable injective module E is hollow,
i.e., every proper submodule is small lin E.
The main results in our report are the following 2 Theorems
: Theorem 1 Let R be a right perfect right weakly QF-3 ring. The
following conditions are equivalent :
(1) R is right hereditary;
(2) e i ,R/e i ,J t is injective for each primitive idempotent e i
and for any integer t;
(3) R is Morita equivalent to direct sum of ring of uppertriangular
matrices over a divisions rings
Theorem 2 Let R be a right perfect ring. If J 2 (R) = 0 or R
is right hereditary then the following conditions are equivalent:
(1) The condition (*) holds;
(2) The condition (**) holds and each e i ,R contains an
unique minimal submodule;
(3) R is right weakly QF-3.
WATER QUALITY AND BREEDING HABITATS
OF ANOPHELLINE MOSQUITO IN NORTH-
WESTERN THAILAND (NO. 796)
Ampornpan K. 1 , Pratap S. 2 , Montip Tiensuwan 3 ., James W.J. 1 ,
and Ratana S. 1
1 Department of Entomology, Armed Forces Research Institute
of Medical Sciences., Bangkok; 2 Department of Tropical
Hygiene, Faculty of Tropical Medicine, Mahidol University,
Bangkok; 3 Department of Mathematics, Faculty of Science,
Mahidol University, Bangkok.
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