6
X
=
JM
for some ideals
I
,
J
of
R
. Thus, (
I
+
J
)
M
=
M
. This implies that
I
+
J
=
R
, by Lemma 0.3 (2). Since
N
M
, we
also have
I
R
. By hypothesis,
J
=
R
this show that
X = M
. Consequently,
M
is co-uniform.
□
Main results
4. Semisimple modules
In this section we study relationships between semisimple module, semisimple ring and their endomorphism
ring. We recall that, an
R
-module
M
is said to be
semisimple module
if every submodule of
M
is a direct summand of
M
and a ring
R
is called
semisimple ring
if
R
is semisimple as an
R
-module, [3]. Before we get our results, we need the
following lemma.
Lemma 4.1
Let
M
be a faithful multiplication
R
-module and
I
,
J
be ideals of
R
. Then
M = IM
JM
if and only if
R = I
J
.
Proof
Suppose that
M = IM
JM
. Then
RM =
IM + JM
= (
I + J
)
M
, by Lemma 0.3 (2), we have
R = I + J
, and
(
I
J
)
M
= 0, which implies that
I
J
= 0 since
M
is a faithful. Therefore,
R = I
J
. Conversely, suppose that
R = I
J
. Since
I
and
J
are ideals of
R
, then
IM
and
JM
are submodules of
M
. Thus,
M = IM + JM
and by Lemma 0.1
(1), we have
IM
JM
= (
I
J
)
M
= 0. Therefore,
M = IM
JM
.
□
We are now ready to state our main theorems of this section.
Theorem 4.2
Let
M
be a faithful multiplication
R
-module over a ring
R
and let
S
= End
R
(
M
). Then the following
statements are equivalent.
(1)
M
is a semisimple
R
-module.
(2)
R
is a semisimple ring.
(3)
S
is a semisimple ring.
(4)
M
is a semisimple
S
-module.
Proof
(1)
(2). Suppose
M
is a semisimple
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
M = IM
N
. Since
M
is a multiplication
R
-module,
N = JM
for
some ideal
J
of
R
. Thus,
M = IM
JM
and by Lemma 4.1 we have
R = I
J
. Therefore,
R
is a semisimple ring.
(2)
(3). Assume that
R
is a semisimple 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 a semisimple
ring, thus
S
is a semisimple ring.
(3)
(4). Suppose
S
is a semisimple ring. Since
M
is a multiplication
R
-module and by Lemma 0.4,
M
is a
faithful multiplication
S
-module. Let
N
be an
S
-submodule of
M
. Then (
N
:
M
) is an ideal of
S
and by assumption,
there is an ideal
J
of
S
such that
S =
(
N
:
M
)
J
. By Lemma 4.1 we have
M =
(
N
:
M
)
M
JM
. Thus,
M = N
JM
.
Therefore,
M
is a semisimple
S
-module.
(4)
(1). Assume
M
is a semisimple
S
-module. Since
M
is a multiplication
R
-module and by Lemma 0.4,
M
is faithful multiplication
S
-module and
M
is finitely generated
S
-module by Lemma 0.5. By Lemma 0.8, End
S
(
M
) is a
semisimple ring.
S
End
S
(
M
) by Lemma 0.6. Since End
S
(
M
) is a semisimple ring, thus
S
is a semisimple ring. Since
M
is
