2
Various properties of multiplication modules are considered. For basic properties of a multiplication module one may
refer to [2], [4] and [10].
In this paper we investigate some further properties and characterizations of faithful multiplication modules
and some related basic results. Moreover, we examine some conditions of semisimple module, semisimple ring and
their endomorphism ring are equivalent and CS-module, CS-ring and their endomorphism ring are equivalent on
multiplication module.
Some characterizations of faithful multiplication modules
We establish some characterizations of faithful multiplication modules. We will begin with the following
lemmas.
Lemma 0.1
[4, Theorem 1.6] Let
M
be a faithful
R
-module. Then
M
is a multiplication module if and only if
(1)
I
M
= (
I
)
M
for any non-empty collection of ideals
I
(
)
of
R
, and
(2) for any submodule
N
of
M
and ideal
A
of R such that
N
AM
there exists an ideal
B
with
B
A
and
N
BM
.
Lemma 0.2
[4, Theorem 2.2] Let
M
(
)
be a collection of
R
-modules. If
M =
M
is a multiplication
module, then
M
is a multiplication module for each
.
Lemma 0.3
[4, Theorem 3.1], Let
M
be a multiplication
R
-module, then the following statements are equivalent.
(1)
M
is finitely generated.
(2) If
A
and
B
are ideals of
R
such that
AM
BM
then
A
B
.
(3)
M
AM
for any proper ideal
A
of
R
.
Lemma 0.4
[6, Lemma 1.10] Let
M
be an
R
-module and
S
= End
R
(
M
). Then if
M
is a multiplication module over a
ring
R
, then
M
is a faithful multiplication module over a ring
S
.
Lemma 0.5
[7, Theorem 2.6] If
M
is a faithful multiplication
R
-module, then
M
is finitely generated.
Lemma 0.6
[8, Theorem 2.4] If
M
is a finitely generated multiplication
R
-module, then End
R
(
M
)
R/
Ann(
M
) as
rings. Moreover, if
M
is faithful then End
R
(
M
)
R
.
Lemma 0.7
[9, Proposition 2.2] If
M
is a multiplication
R
-module, then
M
is a finitely generated as an
R
-module if
and only if
M
is a finitely generated as an
S
-module.
Lemma 0.8
[11, 20.8] For a semisimple
R
-module
M
. Then
M
is finitely generated if and only if End
R
(
M
) is a semisimple
ring.
1. Maximal and minimal submodules
In this section, we give some more information about relationships between maximal ideals of a ring
R
and
maximal submodules of a multiplication
R
-module
M
. Recall that, a submodule
N
of an
R
-module
M
is called
maximal
submodule
of
M
if
N
M
and for any submodule
K
of
M
such that
N
K
(
N
is a submodule of
K
) implies
K
=
M
or
K
=
N
and
N
is called
minimal
submodule
of
M
if
N
0 and for any submodule
K
of
M
such that
K
N
implies
K
= 0
or
K
=
N
, [5].
