7
a faithful multiplication
R
-module and by Lemma 0.5,
M
is finitely generated multiplication
R
-module and hence
S
R
by Lemma 0.6. Since
S
is a semisimple ring, thus
R
is a semisimple ring. Similar with the proof of (3)
(4), we have
M
is a semisimple
R
-module. The proof is complete.
□
Recall that a submodule
N
of
R
-module
M
is called
closed
in
M
if it has no proper essential extension in
M
.
Note that every submodule of
R
-module
M
is a direct summand of
M
if and only if every submodule of
M
is closed,
[5, 139].
Corollary 4.3
Let
M
be a faithful multiplication
R
-module and
N
a submodule of
M
. Then
N
is closed in
M
if and
only if (
N
:
M
) is closed in
R
.
Proof
Let
N
be closed in
M
. Then
N
is a direct summand of
M
and thus
M
is a semisimple module. By Theorem 4.2,
R
is a semisimple ring and we have every ideal of
R
is a direct summand of
R
. Therefore, every ideal of
R
is closed.
According, (
N
:
M
) is closed in
R
. Conversely, assume that (
N
:
M
) is closed in
R
. Then (
N
:
M
) is a direct summand of
R
and hence
R
is a semisimple ring. By Theorem 4.2,
M
is a semisimple module. Therefore,
N
is closed in
M
.
□
5. CS-modules
We closed this section with relationships between CS-module, CS-ring and their endomorphism ring. Recall
that, an
R
-module
M
is called CS
-module
(or
extending module
)
if every submodule of
M
is essential in a direct
summand of
M
. Equivalently,
M
is CS-module if and only if every closed submodule in
M
is a direct summand of
M
.
A ring
R
is called CS
-ring
if
R
is CS
-module
as an
R
-module, [3].
The following theorem could be considerably helpful in the study of the relationship between CS-module,
CS-ring and their endomorphism ring on multiplication module.
Theorem 5.1
Let
M
be a faithful multiplication module over a ring
R
and let
S
= End
R
(
M
). Then the following
statements are equivalent.
(1)
M
is a CS-module as an
R
-module.
(2)
R
is a CS-ring.
(3)
S
is a CS-ring.
(4)
M
is a CS-module as an
S
-module.
Proof
(1)
(2). Suppose that
M
is a CS-module as an
R
-module. Let
I
be an ideal of
R
. Then
IM
is a submodule of
M
and by hypothesis, there is a submodule
N
of
M
such that
IM
e
N
M
. Since
M
is a multiplication module,
N = JM
for some ideal
J
of
R
. Thus,
IM
e
JM
M
. By Lemma 0.2,
JM
is a multiplication module and from Proposition 2.1,
we have
I
e
J
. Next we show that
J
R
. Since
JM
M
, there exists submodule
K
of
M
such that
M = JM
K
.
Since
M
is a multiplication module,
K = AM
for some ideal
A
of
R
. Thus,
M = JM
AM
. By Lemma 4.1,
R = J
A
.
Therefore,
R
is a CS-ring.
(2)
(3). Assume that
R
is a CS-ring. Since
M
is a faithful multiplication
R
-module and by Lemma 0.5,
M
is
finitely generated faithful multiplication
R
-module and hence
S
R
by Lemma 0.6. Since
R
is CS-ring, thus
S
is a CS-ring.
