661
Internat. J. Math. & Math. Sci.
VOL. 17 NO. 4 (1994) 661-666
RELATIVE INJECTIVITY AND CS-MODULES
MAHMOUD AHMED KAMAL
SHAMS UNIVERSITY, FACULTY OF EDUCATION
NATHENATICS DEPAR’INENT, HELIOPOLIS, CAIRO,EGYPT.
(Received August 31, 1992)
ABSTRACT.
In this paper we show that a direct decomposition of modules M
with N homologically independent to the
and only if N is
injective relative to H
As an application,
we prove
N,
inJective hull of H, is a CS-module if
and both of M and N are CS-modules.
that a direct sum of
a non-singular
semisimple
module and a quasi-continuous module with zero socle is quasi-continuous. This
result is known for quasi-inOective modules.
But when we confine ourselves to
CS-modules we need no conditions on their socles.
Then we investigate
direct
sums of CS-modules which are pairwise relatively inective. We show that every
flnite direct sum of such modules is a CS-module. This result is known for
quasi-continuous modules. For the case of infinite direct sums, one has to add
an extra condition.
Finally,
we briefly
direct summands are relatively
KEY WORDS AND PHRASES.
modules in which every two
discuss
inective.
Injective
modules,
self-lnjectlve
rings,
and
generaIization.
1992 AMS SUBJECT CLASSIFICATION CODE.
16A52.
INTRODUCTION.
Let R be a ring and M be a right R-module. The module
M is a CS-module
(for complement submodules are direct summands) provided every submodule of M
is essential in a direct summand of M ,or equivalently, every closed submodule
of M is a direct summand. This is the terminology of Chatters
and
Hajarnavis
[2], one of the first papers to study this concept
Later other terminology, such as extending module, has been used in place
of CS. CS-modules are generalizations of (quasi) continuous modules, which, in
turn, are generalizations of injective
and quasi-injective
modules.
All modules will be unital right modules over a ring R with unit.
A submodule N of a module M is closed in M, if it has no proper essential
extensions in M. X
M and Y
M signify that X is an essential submodule,
and Y is a direct summand, f M. The injective hull of a module M will be
denoted by E(M). A module M is quasi-continuous if it is a CS-module and has
e
e
the following property (C):
3
X
Y
M.
module N of
summand of M.
for
all
X, Y
Ke
M,
with
X
Y
O,
one
has
M is continuous if it is a CS-module and satisfy (C):
if a sub-
M is isomorphic to a
a
2
direct summand of
M, then
N is
direct
M.A. KAMAL
662
For modules M, N and for any f
Hom(M,E(N)), let X =-{ m
M:f(m)
N },
M
N by B
m + f(m)
m
X }. It is clear
that X is an essential submodule of M and that X
B
kerr If -’M (R) N
M
f
f
f
is the projection then
is a monomorphism and X
(B
f
f
LEMMA |. Let M, N be R-modules. Then for every f
Hom(M,E(N)), B and N
and define the submodule B
of
IB
are complements, of each other, in
unique complement of B
PROOF.
M
N
(I)
By
0
N
onto
N
and
f(1), for all
the
essentiality
(It)
#
B
N. If
Hom(N,E(M))
L. Let
and
=
be the natural
B
L, once
To this end, let 0
over N, there
E(N)
f(Ir) N. But
O, then N is
O. Now let L be a submodule
respectively. Then
L
of
(R)
N
O, and that B
M
M
the
N.
(R)
It is clear that
such that L
of
in M
(It)
f(Ir)
Ir
M
of
projections
we
show
that
f)(1) for some
exists
R,
r
N
L.
such
[=(It) + f=(Ir)]
that
N
L= 0
which is a contradiction.
For the second part of the lemma, let Y be a submodule of M
N such that
# 0
O. If
X
then
the
restriction
of
to
f
Y n X provides a nonf
f
f
zero element of Hom(N,E(M)), which contradicts our assumption. Then Y n X
O,
Y
B
f
and thus Y
M
0 (due to
Xf e
Iy
M). It follows that
is a monomorphism,
and thus {Y)
M
0 {otherwise it leads to a contradiction). Therefore Y
N.
LEMMA 2. Let M and N be modules. Then N is M-injectlve if and only if
N
B
N; for every f Hom(M E(N))
f
PROOF.
N is M-inJective if and only if X M, and M
N
B N if and
f
f
M; for every fe Hom(M,E(N)).
REMARK 3. It is known that a module M is quasl-contlnuous if and only if
X
Y, for any submodules X, Y which are complements of each other. An
only if
M
Xf=
immediate consequence of Lemma 1, and Lemma 2,
is that if M
N is quasicontinuous, then M and N are relatively Injective ([I0], Proposition 2.1).
The uniqueness, in the second part of Lemma I,
is used in Proposition 9
to obtain a generalization of the result given in Kamal and Mller [7].
LEMMA 4. {[3], Proposition 1.5) Let A and B be submodules of a module M,
B. If A is closed in B and B is closed in M, then A is closed in M.
with A
COROLLARY 5. Every direct summand of a CS-module is a CS-module.
PROOF. Is obvious.
LEMMA 6. Let M and N be modules, and let A be a submodule of M s N, with
O. Then A is closed in M
A n N
x + f(x)
x
X },
N if and only if A
where X is a closed submodule of X
Hom(M,E(N}). Moreover, if M"
for some f
f
N if and only if A
is uniform, then A is non-zero closed in M
Bf, for some
f
Hom(M,E{N)).
all a
Hom(M,E(N)) such that f(a)
A. Hence A
contained in
Since
in X
f
be the projection of M
Let
PROOF.
exists f
Xf.
x + f(x)
Now if {A)
A is closed in M
e
Xf
then A
N, it follows that Y
e{ Y
+ f(y)
(A);
Xf,
a) for
and thus (A}
Xf.
Y
y
M
N.
and thus (A) is closed
Now if M is uniform, and A is non-zero closed in M
is closed in the uniform module
O, there
(A} }. It is easy to check that (A) is
x
y
N onto M. Since A n N
(1-){a) (i.e. f{a} + (a}
N,
then
Therefore A
0
Bf.
(A)
663
RELATIVE INJECTIVITY AND CS-MODULES
A
Conversely, let
x + f(x)
A _c B
Hom(M,E(N)). It is clear that
f
extension in
X
x
and that A has a proper
if and only if X has a proper essential extension in
Bf
X is closed in X f, it follows that A is closed in B f" But
closed in M e N. Therefore A is closed in M (R) N.
and
Xf,
X is closed in
where
essential
X.
Since
B
by Lemma
is
the major step in studying the property CS for modules
Observe that,
is
the one that deals with the characterization of all closed submodules. So that
Lemma 6 (including its special case, i.e. when M is uniform), can be used in
characterizing CS-modules, which are direct sums of two
modules (see
uniform
Is]).
COROLLARY 7. Let M and N be modules. Then N is M-inOective if and only if
N
O, must have the following form
N, with A
any closed submodule A of M
and f e Hom(M,E(N)}.
M
in
closed
X
is
f(x}
where
+
e
},
X
’x
x
A
M;
PROOF. (). By Lemma 6, and since N is M-inOective if and only if
Xf=
for every f e Hom(M,E(N)).
is a
Let f e Hom(M,E(N)) be an arbitrary element. By Lemma I,
some
for
above
O. Then B has the form
N
closed submodule of M e N with B f
f
g e Hom(M,E(N)), and for some closed submodule Y of M. It follows that,
M is the projection onto M.
M N
(B
Y is closed in M" where
X
Bf
():
f
f
M.
is essential in M, we deduce X
f
COROLLARY 8. Let M be a CS-module, and let N be M-injective. Then every
0 is a direct summand.
N, with A N
closed submodule A of M
Since X
f
PROOF.
Let A be a closed submodule of
A
N
M
(R)
N
with A
N,
O.
Then, by
{x + f(x):x e X}, where X is closed in M and f e Hom(M,E(N)).
It is easy to check that
X (R) M
we have that M
Corollary 7, A
Since
M
is a CS-module,
N- and thus M
X
N
A
M
N.
PROPOSITION 9. Let M and N be modules. Let Hom(N,E(M))
O. Then N is
M-injective and M is a CS-module if and only if every closed submodule A, of
N, with A n N
M
O, is a direct summand.
PROOF.
The necessary condition follows immediately from Corollary
The sufficient condition:
By
Lemma 4, and since
A
N
O,
for
every
closed submodule A of M, M is a CS-module. To show that N is M-injective it is
enough to show M
N, for every f e Hom(M,E(N)). By Lemma I, B is a
Bf
f
N, with B
O; and hence B is a direct summand.
N
f
f
N is the unique complement of B
N, we have that
in M
N
closed submodule of M
Since, by Lemma
f
MeN=B eN.
f
Let M and N be modules. Let Hom(N,E(M))
Theorem 10.
O. Then M
N is a
CS-module if and only if M and N are CS-modules, and N is M-injective.
PROOF. ()
Corollary 5, and Proposition 9.
_
By Proposition 9, it is enough to show that any closed submodule A
of M
N, with A
N
O, is a direct summand. To this end, let A be a
maximal essential extension of A
N in A
By Lemma 4, A is closed in M
N,
O. By Lemma 6 and since Hom(N,E(M))
with A
M
O, it follows that A
N.
()
Since
N is a
A =: A n
CS-module,
M e N
we get that
N
A
N
is a closed submodule of M e N
N is M-injective, it follows, by Corollary 8, that A
a direct summand of N
N.
Thus
with
M
A
A
N
A
A
N
O.
where
Since
Therefore A ls
664
M.A. KAMAL
The following are immediate consequence of Theorem I0.
COROLL 11.
([?], Theorem I}
singular and N is singular. Then
Let M and N be modules, where M is nonN is a CS-module if and only if N is M-
M
injective, and M, N are CS-modules.
COROLLARY 12.
Let M and N be modules, where N is semisimple and M with
zero socle. Then M (R) N is a CS-module if and only if M is a CS-module and N is
M-injective.
PROPOSITION 13.
Let M be a non-singular semisimple R-module, and N be an
R-module, with Soc(N}
O. Then N is quasi-continuous if and only if M
N is
quasi-continuous.
PIOOF.
Let N be quasi-continuous. We show that Hom(N,E(M))
an arbitrary element of Hom(N,E(M)), and let
n
N
there exists an essential right ideal
1’
E(M) is non-singular, it
Since
Ker f
e
N
of R
f(n )I
such that
f(n
follows that
O. Let f be
N. Then, for every
N
O; and thus
0
Ker f.
Hence Kerr has no proper essential extensions in N; i.e. Ker f is closed in N.
Since N is quasi-continuous hence a CS-module we have N
Since
Kerr @ N
Soc(N)
O, it follows that N
O. Then M and N are relatively
O; and thus f
inJective quasi-continuous modules;
M
and therefore
@
N is
quasi-continuous
(see [I0], Corollary 2.14).
In Proposition 13, if M is semisimple but not non-singular or
O, then M m N need not be quasi-continuous. This is illustrated in
14.
Soc(N)
the following examples.
Z. Then,
EXAILE 1. Let M Z/pZ, where p is a prime number; and let N
Soc(N)
N
is not even
as Z-modules, M is singular semisimple and
O, while M
a CS-module (by Corollary 12 ).
o
N
Let M
Let F be a field R
EXAIPLE 2
F
()
Then M is a non-singular simple
R-module, and
continuous R-module, with non-zero socle,
" RR,
show that
This shows that R
O, while I, M
M
)
( )and
N is uniform,
R
where
e RR
M
R
@
hence a
N.
One
[: ):
where
quasi-
can easily
F }.
a
3) e. M ,N is not quasi-continuous.
Let M and N be R-modules, where M is non-singular and N
N is a CS-module if and only if M and N are both CSThen M
does not satisfy (C
PIOPOSITION 15.
is M-injective.
modules.
Let A be a closed submodule of M
POOF. ()
essential extensions in A of A n M and
A
N. Let A and A
N, by Lemma 4. For each a 2
A2, a2 m +
N is essential in A2, there exists an essential
O. Since M
A n N. It follows that ml
such that a
2
N. Since N is a CS-module, and
O; and thus A
deduce m
are closed in M
A
Since
=
2
we get N
A1 @
A
where
A3 a
2
M’@
@
N
A3 =: A
N’=
for some submodule N
N
A2
A n M, we have A
N
O. Since N is
for some submodule
[H’e N*].
O. By Corollary 5,
M"
and
N*
A
N.
of R
right ideal
is non-singular, we
N,
A 2 is closed in
H
I=1
N
over
N
3
Hence A
A3 is closed in
A1
M
that
follows
of M, by Corollary ?
(i =I,Z)
M and n
n; m
of N. By the essentiality of
M-injective, it
It is clear that
M"
be maximal
2
N, respectively. Then
A
i’
with
are CS-modules, where N is M -inJective.
RELATIVE INJECTIVITY AND CS-MODULES
Thus, by Corollary 8, A
.3
{)
eH
N
eH
A
and therefore
665
N.
Is obvious.
REMARK 16. If H is not qon-slngular and N is H-Injective, where both of H
are CS-modules, then H
N need not be a CS-module. This is lllustrated
in Remark 14 (Example 1} by taklng H
Z/pZ
and N
Z.
In Remark 14 (Example 2), we have shown that Soc(N}
0 Is not avoldable
condition for Propositlon 13. This is not the case for CS-modules, as it is
and N
shown in the following.
Let H be a non-slngular semlslmple module. Then H
COROY 17.
N Is a
CS-module for any CS-module N.
THEOREM 18.
Let H
M where the M are M -Injectlve for all
1=1 i’
Then M is a CS-module if and only if M are CS-modules for all i.
PROOF.
If M is a CS-module, then, by Corollary 5, M
is a CS-module for
all i. We show the converse by induction. It is sufficient to prove the result
when n
2. Let M
where the M are CS-modules and M -inJective for
M M
1,2). Let A be a closed submodule of M
j (l,j
1
be a maxlmal essentlal extension of A
M
B2
x + f(x)
f e
O.
follows that X
B
It follows by
M 2 -Injectlve,
x e X }; for some closed submodule X
2
Hom(M2,E(M I)).
where
M Is
Since
B2
M
2
[MI
=: A
inJective. Since M
M2
is a CS-module
Since
M
M]
X2e
M
and hence M
It is clear that
is a CS-module; we have
ffi:
M,
B
M2
B2
BI M*
A
Since
2"
eM
M
2
B2
Thus
O,
2
that
some
and for
2
2
with
Corollary 7,
of M
2
and B
M2,
A
Hence B2 is closed in
In A.
2
B2
Let A
2
XeM 1’ it
2
B 1’
B2
A
and that
M2
is
M1-
(Corollary 8). Then A
is a direct summand of M.
A module M is a DRl-module provided that
submodules of M
any two
are
relatively Inectlve, whenever they form a direct decomposition of M, i.e.
is
M-Inectlve
1,2) whenever M
(i
M M2
From Remark 3, every quasl-contlnuous module is a DRI- module. There are
DRl-modules which are not even CS-modules. In fact every Indecomposable module
is a DRl-module. For an example of a decomposable DRl-module which is not a
R K[x,y]/<x2,xy,y2>. Let S be any
M R S. M is not a CS-module (due
to R Indecomposable and not uniform). Now R, S are relatively Inectlve, and
any two docomposltlons of M are isomorphic (due to R and end(S) local rings);
i. e. M is a DRl-module.
PROPOSITION 19. Every direct summand of a DRl-module is a DRl-module.
PROOF. Is obvious.
PROPOSITION 20. A module M is a quasl-contlnuous module if and only if
K
CS-module, let
simple
be a field, and let
InJectlve R-module,
is a DRI-CS-module.
PROOF.
Let X
Y
=
and consider
M, with X
Y
O. Write M
DRl-module, X is M -inectlve. By Corollary 7, Y
is a closed submodule of
therefore M
X
Y
M*
B.
The converse is obvious.
and f
X
M
Since
M is a
A
where A
a + f(a) :a
Hom(M ,X). By Corollary 5,
M’=
A
B, and
M.A. KAMAL
666
PROPOSITION 21. Let M
El
lel
is a homologically independent family (i.e. Hom(M
M
3
0 for all
E
I)
then M is a DHl-module.
PROOF. Let M
K
K be a decomposition of M. Let ’M
K,
M (iEI) be the canonical pro3ections. Let A =:{aEI: (M
:M
:M
K, and
0}. We show
that K
and hence
(Mj
and hence
K
for all j E I. Now we have K
Mj
9,,(M.).
all j E I. Since the M
.
Therefore K
eaEAM
S =:{ s E I:
(M)
Since
=(M.)
Ee
K
it follows that
(M,)
M
By the same argument we can show that K
0 }. This shows that K and K
([I0], Theorem 2.13)
THEOREM 22.
=e M,
are indecomposables, we have =(M
Let
{Mi:
K"
EI (Mj K)
ejEI(Mj
e M.
for
for all
=
esEs Ms,
where
6
A.
are relatively in3ective.
i E I} be a family of
quasi-
continuous modules. Then the following are equivalent:
I.
2.
i
M
iI
M
M
is quasi-continuous;
is
M -injective for every 3
I.
COROLLARY 23. Let M
M 1’ where the M are quasi-continuous for all
IEI
I. Then M is a DRl-module if and only if M is quasi continuous.
PROOF. Is obvious.
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Rins
Springer Verlage, New York (1973).
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Chatters, A.W. and Ha3arnavis, C.R. Rins in which every Complement RiRht
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3.
Goodearl, K.H.
4.
Harada, M. O__n Modules with
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Harada, M.
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Kamal, M. A.
Theory,
and
Categories
of
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Marcel Dekker, Inc. New York (1976).
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Osaka J. Math. 19(1982),
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Modules in which Complements ar__e Summands, Ph.D. Thesis
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Domains, Osaka
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Kamal, M. A. and Mller, B. J. Th___e Structure o__f Extendin Modules over
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Kamal, M. A. and Mller, B. J.
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Mller, B.J. and Rizvi, S.T. O__n In,]ective and Quasi-continuous Modules,
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Mller, B.J. and Rizvi, S.T. Direct Sum of Indecomposable Modules, Osaka
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