Contemporary Mathematics
Volume 419, 2006
ON COUNTABLY I:-CS MODULES
ADEL N. ALAHMADI, HUSAIN S. AL-HAZMI, AND
PEDRO A. GUlL ASENSIO
ABSTRACT. We study a criterion for a uniform countably
~-CS
module to be
(~-)quasi-injective.
As a consequence, we get necessary and sufficient condi-
tions that force a direct sum of indecomposable modules to be
~-CS
provided
that it is countable
~-CS.
1.
INTRODUCTION
A module
M
over a ring
R
is called CS (or
extending,
see [3]) if every sub module
is essential in a direct summand of
M.
And it is called (countably) I:-CS module
if every direct sum of (countably many) copies of M is CS. The ring R is called
right I:-CS if it I:-CS as right R-module. Right I:-CS rings were first studied
by Oshiro under the name of co-H-rings [11]. He proved that every right I:-CS
ring is both-sided artinian and therefore, it is a direct sum of uniform right ideals.
After these results, the problem of whether a I:-CS module is also a direct sum
of uniform submodules became a major problem for these modules. This question
was positively answered in [6, 8, 7], where it is also shown that any module
M
such
that
M(I)
is CS for some uncountable index set
I,
is actually I:-CS. This is the
best bound possible, since there exist examples of non-singular right self-injective
rings R such that RJ:o) is CS, but RR is neither I:-CS nor a direct sum of uniform
ideals (see [3, Example 12.20 (i)]).
However, there are not known examples of countably I:-CS modules or rings
that are a direct sum of uniform submodules but they are not I:-CS. This led
Huynh and Rizvi to ask in [9] whether a countably I:-CS ring (or module) that is
a direct sum of uniforms might be I:-CS. After the results in [1] (see also [4, 5]),
this problem is equivalent to ask whether a uniform countably I:-CS module must
be (I:-)quasi-injective.
In this paper, we study necessary and sufficient conditions that force a direct
sum
tB
1
Mi
of indecomposable modules to be I:-CS provided that it is countably I:-
CS. Our main result states that this is the case if and only if every
Mi
has w1-ACC
on monomorphisms (see next section for the definition). Equivalently,
if
and only
if the quasi-injective hull of each
Mi
has w1-ACC on submodules isomorphic to
Mi.
Throughout this paper all rings R will be associative and with identity. And
Mod-R will denote the category of right R-modules. By a module we will mean
a unital right R-module. A submodule
C
of a module
M
is said to be
closed
in
M
if it has no proper essential extension in
M.
A submodule
X
of
M
is called a
2000 Mathematics Subject Classification. 16D50, 16D90, 16L30.
Key words and phrases. Quasi-injective modules, uniform modules,
~-CS
Modules.
The third author has been partially supported by the DGI (BFM2003-07569-C02-01, Spain)
and by the Fundaci6n Seneca (PI-76/00515/FS/01).
@2006 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/419/07991
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