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