Advances in Pure Mathematics, 2011, 1, 359-362
doi:10.4236/apm.2011.16064 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Some Equivalences and Dualities via Static Modules
Salah Al-Nofayee1, Syed Khalid Nauman2
1Department of Mathematics, Taif University, Taif, Saudi Arabia
2Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Received July 14, 201 1; revised August 16, 2011; accepted August 25, 2011
Abstract
For a ring A, an extension ring B, a fixed right A-module M, the endomorphism ring D formed by M, the
endomorphism ring E formed by A
M
B, and the endomorphism ring F formed by , we
present equivalences and dualities between subcategories of B-modules which are finitely cogenerated injec-
tive as A-modules and E-modules and F-modules which are finitely generated projective as D-modules.

,
A
HomB M

Keywords: Static Modules, Finitely Cogenerated Injective Modules, Finitely Generated Projective Modules
1. Introduction
Let A be any ring, M a fixed right A-module and
. An object V in Mod-A (respectively W
in Mod-D) is said to be M-static (respectively M-adstatic),
in case V remains invariant under the composite covari-
ant functor

A
DEndM

,
AD
H
om

,
A
M M

(respectively
D
H
omM  M. We denote by Stat(M) and Adst(M)
the classes of all static and adstatic objects of Mod-A and
Mod-D, respectively. We will use the notation FCI-A and
FGP-D (D-FGP) for the classes of all finitely cogener-
ated injective and finitely generated projective objects in
Mod-A, Mod-D (D-Mod) respectively.
It is clear that
 
,: :
AD
H
omMStat MAdstat MM
is an equivalence and a special case of this equivalence is
an equivalence between Mod(A:weak M) and Mod(D:
weak D) where Mod(A:weak M) is the full additive sub-
category of all those objects which weakly divide (i.e.
divide some finite direct sum of copies of) M in Mod-A
and Mod(D:weak D) = FGP-D.
In [1], which is an extension of the work of Xue in [2],
it is proved that M is a finitely cogenerated injective co-
generato r o f Mod-A iff Mod(A:weak M) = FCI-A and that
this fact is equivalent to the existence of an equivalence
or a duality between FCI-A and FGP-D or FCI-A and
D-FGP, respectively.
Let B be another ring and α:A→B a ring homomorph-
ism. Suppose that


BA
EEndM B
:DE
. Then the ring
homomorphism


 
m b
defined via
, for all (d,m,b)∈D × M × B
is clearly identity preserving. Similarly if
 
dmb  d




,
BA
F
EndHomBM
:DF
, then the ring homomorphism

 defined via


d
f
df  , for all d∈D
and


,
A
f
HomB M, is clearly identity preserving.
Let us set
Mod(B:weak M) = {V∈Mod-B: weakly divides M
in Mod-A},
A
V
Mod(B:FCI-A) = {V∈Mod-B: VA∈FCI-A },
Mod(E:FGP-D) = {W∈Mod-E: WD∈FGP-D},
Mod(E:D-FGP) = {W∈E-Mod: WD∈D-FGP},
Mod(F:FGP-D) = {W∈Mod-F: WD∈FGP-D},
Mod(F:D-FGP) = {W∈F-Mod: WD∈D-FGP}.
With the assumption that A
M
B is M-static in
Mod-A, an equivalence between subcategories Mod(B:
weak M) and Mod(E:FGP-D) of Mod-B and Mod-E, re-
spectively, is established in [3]. This in fact is a gener-
alization of the work of Cline [4] and Dade [5] on stable
Clifford theory. In this work using the same assumption
it is proved that Mod(B:FCI-A) and Mod(E:FGP-D) are
equivalent and with some additional assumption dualities
between Mod (B:FCI-A) and both Mod(F:D-FGP) and
Mod(E:D-FGP) are deduced.
We assume that the rings are associative with iden tity,
the ring homomorphisms are identity preserving , all (left,
right) modules are unital, and all subcategories are full
and additive.
2. Equivalences and Dualities
We fix here all the notations and terminology from the
previous sectio n.
The following Theorem is proved in [1], Theorem 3.

S. AL-NOFAYEE ET AL. 360
Theorem 2.1 The following statements are equivalent
for a right A-module M.
1) M is a finitely cogenerated injective cogenerator in
Mod-A,
2) Mod-(A:weak M) = FCI-A,
3)

,: :
AD
H
omMFCIAFGP DM  de-
fine an equivalence,
4)



,:: ,
AD
H
omMFCIADFGPHomM 
define a duality.
Corollary 2.2 Let M be a right A-module. If M is a
finitely cogenerated injective cogenerator in Mod-A, then
Mod(B:weak M) = Mod(B:FCI-A).
Proof. By Theorem 2.1-(2) and the definitions of the
subcategories Mod(B:FCI-A) and Mod(B:weak M), the
result follows. □
Remark 2.3 From now on we assume that A
M
B

is M-static as an A-module. With this assumption, one
can deduce that AD
M
BE M in D-Mod-A. The
details of internal maps of this isomorphism and their
proofs can be seen in [3].
The following Theorem is proved in [3], Theo rem 5.5.
Theorem 2.4 For a right A-module M, the restrictions
of the additive functors
 
,and
BA EA

H
om MBMB
form an equivalence of the full additive subcategories
Mod(B:weak M) and Mod(E:FGP-D) of Mod-B and
Mod-E, respectively.
Proposition 2.5 For a right A-module M, the follow-
ing statements are equivalent.
1) Mod(B:weak M) = Mod(B:FCI-A),
2) The restrictions of the additive functors



, and
BA EA
H
omMB MB form an equi-
valence of the full additive subcategories Mod(B:FCI-A)
and Mod(E:FGP-D) of Mod-B and Mod-E, respectively.
Proof. (1)⇒(2) By Theorem 2.4.
(2)⇒(1) By Theorem 2.4, it is clear that




:
:Im
:
EA
ModEFGPD
ModBweak MMB
Mod B FCIA

 

Remark 2.6 Recall that and

BA
EEndM B
 
,
BA
F
EndHomB M
1) For the following one can see [6], Lemma 3.2. We
have
 
,,
AA
H
omHomBMMF,
and in this case
 


,,, ,
DAA D
H
omHom Hom BM M MHomFM
So if

,
A

H
omB M is M-reflexive, then
 
,,
AD

H
omB MHomFM,
therefore





 
,, ,,
FA FD
HomW HomBMHomW HomFM.
But by the adjoint associativity theorem we have



 
,, ,
FD D
H
omWHomFMHomW M
thus



 
,, ,
FA D
H
omWHomB MHomWM


,
AA

M
BHomBM
2) Suppose that as B-mo-
dules and hence as A-modules. Then we have the follow-
ing sequence of D-isomorphisms:

 


 
 
 
,
,,
,
,.
BA A
BA A
AAB
AA
EHomMB MB
HomMBHomB M
HomMBBM
HomMB M





This means that
  


,,
DDAA
,
H
om EMHom HomMBMM


A
M
B in E-Mod-A. Further, if is M-reflexive, then


,
DAD
H
omEMMBEM 
in E-Mod-A.
3) On the other hand, according to the assumption
that




,
AA
M
BHomBM
F
, it is easy to see that
E

in D-Mod.
The following Proposition is proved in [6, Proposition
3.3].
Proposition 2.7 Let M be a right A-module. If


,-
A
H
omBMreflexive is M-reflexive, then
1)
B
V


,-
A
is
H
omBMreflexive if and only if
is M-reflexive.
A
V
2) F is W


,-
A
H
omBMreflexive if and only if
is M-reflexive.
D
W
Theorem 2.8 Let M be a right A-module. Let


,
A
H
omB M be M-reflexive. Then the restrictions of
the additive functors
 
,,
BA

H
omHomBM and




,
FA
,
H
om HomB M form a duality of the full addi-
tive subcategories Mod(B:weak M) and Mod(F:D-FGP)
of Mod-B and F-Mod, respectively.
Proof. Let V∈Mod(B:weak M), hence V∈Mod(A:
weak M). This means that n
M
VU, for some U and
some positive integer n. Now we have

***nn
DM VU,
 
*,
A
H
om M 


*,
A
VHomVM where . Thus
∈D-FGP. We have the following isomorph isms,



  

,, ,
,
BA AB
A
H
omVHomB MHomVBM
HomVM



S. AL-NOFAYEE ET AL.361
   
,,
BA
H
omVHomB M
n
DWQ
Hence ∈D-FGP and so be-
longs to Mod(F:D-FGP). Let W∈Mod(F:D-FGP). Then
W∈D-FGP and so , for some Q in D-Mod
and some positive integer n. Now we h ave
where the fourth isomorphism is due to (2) and the fifth
isomorphism is due to the fact that V is

*
** nn
WQ DM,
where
 
*,
D
H
om M 

,
D
HomW M
. Th erefore
weakly divides M in Mod-A. i.e.
∈Mod(A:weak M) and hence it belongs to Mod(B:
weak M). Now as we can see from Remark 2.6-(1),
*
W
*
W



,, ,
FA D
H
omWHomB MHomWM
We deduce that
 
,,
FA

H
omWHomB M∈Mod(B:
weak M).
According to the fact that Mⁿ and Dⁿ are M-reflexive,
it is clear that for every V in Mod(A:weak M), V is
M-reflexive and for every W in D-FGP, W is M-reflexive.
Applying Proposition 2.7, since

,
A

H
omB Mis M-
reflexive, every V in Mod(B:weak M) = Mod(B:FCI-A) is
, and every W in Mo d(F:D-FGP)
is . □

,-reflexiv
A
HomB M

,-refle
A
HomB M
e
e

xiv
Theorem 2.9 Let M be a right A-module. Let

,
A
H
omB M be M-reflexive. Then the following state-
ments are equivalent for M.
1) Mod(B:weak M) = Mod(B:FCI-A),
2) The restrictions of the additive functors
 
,,
BA

H
omHomBMand
 
,,
FA

H
omHomB M
form a duality of the full additive subcategories Mod
(B:FCI-A) and Mod(F:D-FGP) of Mod-B and F-Mod,
respectively.
Proof. (1)⇒(2) By Theorem 2.8.
(2)⇒(1) Consider the following isomorphisms ob-
tained by the adjoint associativity theorem and Remark
2.6, we have
  
,, ,
BA A
H
om VHomBMHom VM
and
  
,, ,
FA D

H
omWHomB MHomWM (2)
Let V∈Mod(B:FCI-A), then
∈Mod(F:D-FGP).
 
*,,
BA
VHomVHomBM*n
DV Q
 
So , for some Q and some positive in-
teger n. Now we have the following sequence of iso-
morphisms,








*
*
*
,
,
,,
,, ,
,,
nn
D
D
DD
FA D
D
MHomDM
HomVQ M
HomVMHomQ M
H
omVHom BMHom QM
VHomQM







,-reflexiv
A
HomB Me. So V∈Mod(A:weak M) and
therefor it belongs to Mod(B:weak M).
Conversely suppose that V∈Mod(B:weak M). This
means that n
M
VU

, for some U and some positive
integer n. Now we have the following isomorphisms,



 
,
,
,,
nn
A
A
AA
DHomMM
HomVU M

H
omV MHomU M



Thus


,
A
H
omV M∈D-FGP. By (1) it is clear that




*,,
BA
VHomVHomBM∈Mod(F:D-FGP).
Therefore by the given d ua lity




*,,
FA
H
omVHomBM∈Mod (B:FCI-A). (3)
Now since V∈Mod(A:weak M), it is M-reflexive, so
by Proposition 2.7 it is


,-reflexiv
A
HomB Me. Hence
the result follows by (3). □
Theorem 2.10 Let M be a right A-module. Let




,
AA
M
BHomBM as B-modules and let
A
M
B

be M-reflexive. Then the following statements
are equivalent for M.
1) Mod(B weak M) = Mod(B:FCI-A),
2) The restrictions of the additive functors


,
BA
Hom MB



EA

M
B  and form an equi-
valence of the full additive subcategories Mod(B:FCI-A)
and Mod(E:FGP-D) of Mod-B and Mod-E, respectively.
3) The restrictions of the additive functors


,
BA
H
omM B and

,
EA

H
omM B form a
duality of the full additive subcategories Mod(B:FCI-A)
and Mod(E:D-FGP) of Mod-B and E-Mod, respectively.
Proof.
(1)⇔(2) By Proposition 2.5.
(1)⇔(3) With the assumption


,
AA

M
BHomBM ,
This is clear from Theorem 2.9 (see Remark 2.6-(2)-
(3)). □
Corollary 2.11 Let M be a right A-module such that M
is a finitely cogenerated injective cogenerator in Mod-A.
Let




,
AA
M
BHomBM as B-modules and let
A
M
B

be M-reflexive. Then the restrictions of the
additive functors


,
BA
H
om MB and


,
EA
H
omM B form a duality of the full additive
subcategories Mod(B:FCI-A) and Mod(E:D-FGP) of
Mod-B and E-Mod, respectively.
Corollary 2.12 Let M be a right A-module. Let


A
M
BM

 
,
AA
H
omB MMB and as D-B-
bimodules. Then the following statements are equivalent
for M.

S. AL-NOFAYEE ET AL. 362

1) Mod(B:weak M) = Mod(B:FCI-A),
2) The restrictions of the additive functors
and

,
BA
Hom MB

EA
M
B  form an
equivalence of the full additive subcategories Mod(B:
FCI-A) and Mod(E:FGP- D) of Mod-B and Mod-E, re-
spectively.
3) The restrictions of the additive functors

,
BA
H
omM B and

,
EA

H
omM B form a
duality of the full additive subcategories Mod(B:FCI-A)
and Mod(E:D-FGP) of Mod-B and E-Mod, respectively.
Proof. Since A

,
A

H
omB MMB

,-reflexive
A
HomB M
xive
as D-B-bi-
modules, V is if and only if V
is A. The assumption -reMBfle

A
M
BM,
implies the fact that

A
M
B is M-reflexive. Hence
the proof follows by Theorem 2.10. □
Corollary 2.13 Let M be a right A-module such that M
is a finitely cogenerated injective cogenerator in Mod-A.
Let

A
M
BM and

,
AA

H
omB MMB

A
as
D-B-bimodules. Then the restrictions of the additive
functors ,
B
H
om MB,
EA
and

H
om MB
form a duality of the full additive subcategories Mod(B:
FCI-A) and Mod(E:D-FGP) of Mod-B and E-Mod, re-
spectively.
3. Acknowledgements
This work is supported by Taif University under Project
Number (1-430-368).
4. References
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