3
Proposition 1.1
Let
M
be a faithful multiplication
R
-module. Then a submodule
N
of
M
is maximal if and only if
there exists maximal ideal
I
of
R
such that
N
=
IM
.
Proof
Suppose that
N
is a maximal submodule of
M
. Then there exists an ideal
I
of
R
such that
N
=
IM
. It is
sufficient to prove that
I
is a maximal ideal of
R
. For any ideal
J
of
R
such that
I
J
R
. Thus,
N
=
IM
JM
M
.
Since
N
is a maximal submodule of
M
, either
JM
=
IM
or
JM
=
M
. If
JM
=
IM
then
J
=
I
by Lemma 0.3 (2). If
JM
=
M
then
J
=
R
, again by Lemma 0.3 (2). Therefore,
I
is a maximal ideal of
R
. Conversely, suppose that
N
=
IM
for some
maximal ideal
I
of
R
. Let
X
be a submodule of
M
such that
N
X
M
. Thus,
I
(
X
:
M
)
R
. Since
I
is maximal
ideal of
R
, either (
X
:
M
) =
I
or (
X
:
M
) =
R
If (
X
:
M
) =
I
then
X
= (
X
:
M
)
M
=
N
. If (
X
:
M
) =
R
then
X
= (
X
:
M
)
M
=
M
.
This show that
N
is a maximal submodule of
M
.
□
Corollary 1.2
Let
M
be a faithful multiplication
R
-module and
N
a submodule of
M
. Then
N
is a maximal submodule
of
M
if and only if (
N
:
M
) is a maximal ideal of
R
.
In the next proposition, we show some results which are dually properties of maximal submodules of
multiplication modules.
Proposition 1.3
Let
M
be a faithful multiplication
R
-module. Then a submodule
N
of
M
is minimal if and only if
there exists minimal ideal
I
of
R
such that
N
=
IM
.
Proof
Suppose that
N
is a minimal submodule of
M
. Then there exists an ideal
I
of
R
such that
N
=
IM
. It is
sufficient to prove that
I
is a minimal ideal of
R
. For any ideal
J
of
R
such that
J
I
,
JM
IM
and by hypothesis,
either
JM
= 0 or
JM
=
IM
. If
JM
= 0 then
J
= 0 because
M
is a faithful. If
JM
=
IM
then
J
=
I
, by Lemma 0.3 (2).
Therefore,
I
is a minimal ideal of
R
. Conversely, assume that
N
=
IM
for some minimal ideal
I
of
R
. Let
X
be a
submodule of
M
such that
X
N
. Thus, (
X
:
M
)
I
. By assumption, either (
X
:
M
) = 0 or (
X
:
M
) =
I
. If (
X
:
M
) = 0
then
X
= 0. If (
X
:
M
) =
I
then
X
= (
X
:
M
)
M
=
N
. This show that
N
is a minimal submodule of
M
.
□
Corollary 1.4
Let
M
be a faithful multiplication
R
-module and
N
a submodule of
M
. Then
N
is a minimal submodule
of
M
if and only if (
N
:
M
) is a minimal ideal of
R
.
2.
Essential and co-essential submodules
In this section we will be concerned with relationships between essential respectively. co-essential ideals of a ring
R
and essential resp. co-essential submodules of a multiplication
R
-module
M
. Recall that, a submodule
N
of
R
-module
M
is
called
essential
in
M
,
N
e
M,
if for each submodule
X
of
M
,
N
X
= 0 implies
X
= 0. If
N
is essential in
M
, we say that
M
is an
essential extension
of
N
. Dually,
N
is called
co-essential
in
M
, if for each submodule
X
of
M
,
N+X = M
implies
X = M
.
An ideal
I
of
R
is called
essential
resp.
co-essential
in
R
if
I
is an essential resp. co-essential in
R
R
as an
R
-module [5].
Proposition 2.1
Let
M
be a faithful multiplication
R
-module. Then a submodule
N
of
M
is essential in
M
if and only
if there exists essential ideal
I
of
R
such that
N
=
IM
.
Proof
Suppose that
N
is an essential submodule of
M
. Then there exists an ideal
I
of
R
such that
N
=
IM
. It is
sufficient to prove that
I
is an essential ideal of
R
. For any ideal
J
of
R
such that
I
J
= 0. Let
J
0 we have
JM
is non-
zero submodule of
M
. Thus,
N
JM
= 0 by Lemma 0.1 (1). Since
N
is an essential submodule of
M
, then
JM
= 0, a
contradiction. Thus
J
= 0 and we obtain
I
is an essential ideal of
R
. Conversely, suppose that
N
=
IM
for some essential
