supplement submodules of injective modules
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Supplement Submodules of Injective ModulesJohn Clark a , Derya Keskin Tütüncü b & Rachid Tribak ca Department of Mathematics and Statistics, Otago University, Dunedin, New Zealandb Department of Mathematics, Hacettepe University, Beytepe, Ankara, Turkeyc Département de Mathématiques, Faculté des Sciences de Tétouan, Tétouan, MoroccoPublished online: 22 Nov 2011.
To cite this article: John Clark , Derya Keskin Tütüncü & Rachid Tribak (2011): Supplement Submodules of Injective Modules,Communications in Algebra, 39:11, 4390-4402
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Communications in Algebra®, 39: 4390–4402, 2011Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2010.524464
SUPPLEMENT SUBMODULES OF INJECTIVE MODULES
John Clark1, Derya Keskin Tütüncü2, and Rachid Tribak3
1Department of Mathematics and Statistics, Otago University, Dunedin,New Zealand2Department of Mathematics, Hacettepe University, Beytepe, Ankara, Turkey3Département de Mathématiques, Faculté des Sciences de Tétouan,Tétouan, Morocco
An R-module M is called almost injective if M is a supplement submodule of everymodule which contains M . The module M is called F -almost injective if every factormodule of M is almost injective. It is shown that a ring R is a right H-ring if andonly if R is right perfect and every almost injective module is injective. We prove thata ring R is semisimple if and only if the R-module RR is F -almost injective.
Key Words: Almost injective modules; F -almost injective modules; Injective modules; Liftingmodules; Noncosingular modules; Small modules; Weakly injective modules.
2000 Mathematics Subject Classification: 16D50; 16D99; 16L30.
1. INTRODUCTION
All rings are associative with identity element and all modules are unitaryright modules. Let M be any module and let S and K be submodules of M . Theinjective hull of M will be denoted by E�M�. The notation S � M means that Sis a small submodule of M . The submodule K is called coclosed in M if for everysubmodule A of M with A ⊆ K, K/A � M/A implies K = A or, equivalently, givenany proper submodule L of K, there is a submodule N of M for which K + N = Mbut L+ N �= M . A submodule N of M is called a supplement of K in M if N isminimal with respect to the property M = K + N , equivalently, M = K + N and K ∩N � N . A submodule L of M is called a supplement submodule in M provided thereexists a submodule X of M such that L is a supplement of X in M . All supplementsubmodules are coclosed (see, for example, [5, 20.2]). If every submodule of M hasa supplement in M , then M is called supplemented. The module M is called weaklysupplemented if for every submodule N of M , there exists a submodule K of M suchthat M = N + K and N ∩ K � M . The module M is called amply supplemented if forany two submodules A and B of M with M = A+ B, there exists a supplement P ofA in M such that P is contained in B.
Received April 19, 2010; Revised August 18, 2010. Communicated by T. Albu.Address correspondence to Prof. Derya Keskin Tütüncü, Department of Mathematics, Hacettepe
University, Beytepe 06800, Ankara, Turkey; E-mail: [email protected]
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A module M is called lifting (or satisfies �D1�) if for every submodule X ofM , there exists a direct summand K of M such that K ≤ X and X/K � M/K,equivalently, for every submodule X of M there exist submodules M1 and M2 of Msuch that M = M1 ⊕M2, M1 ≤ X and X ∩M2 � M2. By [5, 22.3], M is lifting if andonly if M is amply supplemented and every coclosed (supplement) submodule of Mis a direct summand of M . Following [20], the module M is called noncosingular iffor every nonzero module N and every nonzero homomorphism f � M → N , Im f isnot a small submodule of N .
In this article, we introduce the notion of almost injective modules. A moduleM is said to be almost injective if M is a supplement submodule of every modulewhich contains M . The module M is called F -almost injective if every factor moduleof M is almost injective. Of course, every injective module is almost injective.The notion of almost injective modules is one of many possible generalizationsof injective modules. In [23], the author introduced another such generalization,namely, weakly injective modules. A module M is called weakly injective if M is acoclosed submodule of every extension of it. We recall that a module M is calledsmall if it is a small submodule of some module; otherwise M is called nonsmall.By [14, Theorem 1], M is small if and only if M is small in its injective hull. InProposition 2.2 we note that a hollow module H is weakly injective if and only ifH is almost injective if and only if H is nonsmall. Since supplement submodules arecoclosed, the following implications hold:
injective ⇒ almost injective ⇒ weakly injective.
In Section 2, we provide some examples to separate these three properties.In Section 3, we investigate some properties of almost injective modules. We
show that a finite direct sum M1 ⊕ · · · ⊕Mn is almost injective if and only if eachMi�1 ≤ i ≤ n� is almost injective (Theorem 3.5). We also prove that a ring R is aright H-ring if and only if R is right perfect and every almost injective module isinjective (Corollary 3.12).
In Section 4, we study F -almost injective modules. After constructing anexample which shows that, in general, an almost injective module need not be F -almost injective (Example 4.2), we prove that for a ring R, the module RR is F -almost injective if and only if R is a semisimple ring (Proposition 4.4). Then weestablish some characterizations of rings whose almost injective modules are F -almost injective. It is shown that this class of rings contains right hereditary rings(Example 4.3).
Section 5 deals with lifting F -almost injective modules. We prove that anylifting F -almost injective module is a direct sum of hollow modules (Corollary5.4). We also prove that over any commutative ring, a local module L is F -almostinjective if and only if L is simple injective (Theorem 5.6).
2. EXAMPLES
We begin with two results where the almost injective and weakly injectiveproperties coincide.
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Proposition 2.1. Let R be a right perfect ring. Then an R-module is almost injectiveif and only if it is weakly injective.
Proof. Since R is right perfect, every R-module M is supplemented,and so its supplement submodules and coclosed submodules coincide.(See [5, 20.3 and 27.24].) �
Proposition 2.2. Consider the following statements for a nonzero module H:
(i) H is almost injective;(ii) H is weakly injective;(iii) H is nonsmall.
Then �i� ⇒ �ii� ⇒ �iii� and, if H is hollow, the statements are equivalent.
Proof. We have already noted that (i) ⇒ (ii).For (ii) ⇒ (iii), suppose that H is a proper submodule of a module M . By (ii),
H is coclosed in M and so, for any proper submodule K of H , there is an N ≤ Mwith H + N = M but K + N �= M . It follows that N �= M , and so H is not smallin M .
Now suppose that H is hollow and that (iii) holds. Let M be any module whichcontains H . Since H is not small in M , there is a proper submodule L of M suchthat M = H + L. On the other hand, we have L ∩H �= H . Thus L ∩H � H , and soH is a supplement of L in M , proving (i). �
Example 2.3. If R is a right perfect local ring, then the module RR is almostinjective.
Proof. By [9, Corollary 2.5], the R-module RR is nonsmall, and so we may applyProposition 2.2. �
Lemma 2.4. Let E be an injective module. If M is minimal (with respect to inclusion)among the nonsmall submodules in E, then M is hollow.
Proof. It is clear that every proper submodule of M is small in E. Then M = N +K implies that N = M or K = M since otherwise N � E and K � E, giving M =N + K � E, a contradiction. Therefore, M is hollow. �
Example 2.5. Let R be a commutative noetherian ring, and let M1, M2� � � � �Mn
be maximal ideals of R. By [16, Proposition 3], the R-module E = ⊕ni=1 E�R/Mi� is
artinian injective. Let H be a module minimal among nonsmall submodules in E.Then H is almost injective by Proposition 2.2 and Lemma 2.4.
An R-module M is called divisible if, for every s ∈ R which is not a zero divisorand every x ∈ M , there exists y ∈ M with x = ys. Every injective right R-module isdivisible, by [15, Theorem 3.1].
Example 2.6. Let R be a Dedekind domain, and let M be an almost injective R-module. Then, since M is a supplement submodule of its injective hull E�M�, there
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is a submodule N of E�M� such that M is a supplement of N . Therefore, for everynonzero element r ∈ R, we have Mr + N = E�M�. Then, by the minimality of M , weget Mr = M . Thus M is a divisible module, and so, since R is a Dedekind domain,M is injective. This has shown that every almost injective R-module is injective.
Although every injective module is almost injective, the following exampleshows that there are also almost injective modules which are not injective.
Example 2.7. Let R be a right artinian local ring which is not quasi-Frobenius.Since R is a right perfect ring, the right R-module RR is almost injective by Example2.3. On the other hand, R is not right self-injective by [13, Theorem 15.1].
(1) The ring R constructed in [13, Exercise 2, p. 454] is local artinian but not quasi-Frobenius.
(2) Let R be a commutative local artinian ring such that R is not a principal idealring. By [19, Theorem 6.7], there exists an ideal I of R such that the ring R/Iis not self-injective. Thus the ring R/I is local artinian but not quasi-Frobenius.For a particular example, we can take R to be the ring k�x2� x3�/�x4�, where k isa field (see [2, Example on p. 91]).
Recall that a ring R is called a right V -ring if every simple right R-module isinjective or, equivalently, Rad�M� = 0 for every right R-module M (see [13, Theorem3.75]). The following characterization is given in [7, Proposition 2.1].
Proposition 2.8. The ring R is a right V -ring if and only if every R-module is weaklyinjective.
By analogy, we prove the following proposition.
Proposition 2.9. The ring R is semisimple if and only if every R-module is almostinjective.
Proof. The necessity is clear.If all R-modules are almost injective, so weakly injective, then, by Proposition
2.8, R is a right V -ring. Let M be any R-module, and let N be a submodule of M .Then there is a submodule L ≤ M such that N is a supplement of L in M . ThusL ∩ N � N . However Rad�N� = 0 and so L ∩ N = 0, whence M = L⊕ N . It followsthat M is semisimple and so R is semisimple. �
It follows from Propositions 2.8 and 2.9 that, over any right V -ring which isnot semisimple, there exist weakly injective modules which are not almost injective(see the following example).
Example 2.10. Let k be a field and R = �i=1ki with ki = k for each i ∈ �. Then R
is a commutative von Neumann regular ring by [10, p. 264] and so a right V -ring by[13, Corollary 3.73]. Moreover, the ideal A = ⊕
i=1ki is not a direct summand of RR.Indeed A cannot be a supplement in RR since otherwise there is a submodule
B of R such that A+ B = R and A ∩ B � A. Then, since Rad�A� = 0, we have A⊕
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B = R, a contradiction. Thus A is not almost injective. On the other hand, A isweakly injective by Proposition 2.8.
It is easy to see that each ki is a simple R-module and so (almost) injective.Thus this example shows also that the class of almost injective modules is not alwaysclosed under infinite direct sums.
3. SOME PROPERTIES OF ALMOST INJECTIVE MODULES
For the following result see, for example, [5, 3.7(6) and 20.6].
Lemma 3.1. Let F ≤ E ≤ M be submodules.
(1) If F is (coclosed) a supplement in M , then F is (coclosed) a supplement in E.(2) If E is (coclosed) a supplement in M , the following are equivalent:
(a) F is (coclosed) a supplement in E;(b) F is (coclosed) a supplement in M .
Proposition 3.2. The following are equivalent for an R-module M:
(i) M is almost (weakly) injective;(ii) There is an injective module E containing M such that M is (coclosed) a
supplement in E;(iii) M is (coclosed) a supplement in its injective hull E�M�.
Proof. (i) ⇒ (ii) is obvious.
(ii) ⇒ (iii) Let X be an injective module such that M ≤ X and M is (coclosed)a supplement in X. Since E�M� is a direct summand of X, M is (coclosed) asupplement in E�M� by Lemma 3.1(1).
(iii) ⇒ (i) Let Y be a module such that M ≤ Y . Then M ≤ E�Y�. But E�M� isa direct summand of E�Y�. Thus M is (coclosed) a supplement in E�Y� by Lemma3.1(2). Therefore, M is (coclosed) a supplement in Y by Lemma 3.1(1). �
The above proposition allows us to conclude that the classes of almostinjective modules and supplement submodules of injective modules coincide.
Corollary 3.3. Every supplement submodule of an almost injective module is almostinjective.
Proof. Let M be an almost injective module, and let N be a supplement submoduleof M . Since M is a supplement in E�M�, N is a supplement in E�M� by Lemma3.1(2). Thus N is almost injective by Proposition 3.2. �
Lemma 3.4. Let M be a module and let N be a submodule of M . If N and M/N arealmost injective, then so is M .
Proof. By [5, 20.5(2)]. �
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Theorem 3.5. A finite direct sum M1 ⊕ · · · ⊕Mn is almost injective if and only if Mi
is almost injective for each i = 1� � � � � n.
Proof. By Corollary 3.3 and Lemma 3.4. �
Corollary 3.6. A finite direct sum H1 ⊕ · · · ⊕Hn of hollow modules Hi is almostinjective if and only if each Hi is a nonsmall module.
Proof. This follows from Theorem 3.5 and Proposition 2.2. �
Corollary 3.7. Let R be a commutative noetherian local ring with maximal ideal Jand M = H1 ⊕ · · · ⊕Hn be a finite direct sum of local R-modules Hi. Then M is almostinjective if and only if AnnR�J� �⊆ AnnR�Hi� for each i = 1� � � � � n.
Proof. By Theorem 3.5, M is almost injective if and only if Hi is almost injectivefor each i and Proposition 2.2 shows that this is so if and only if each Hi is weaklyinjective. This in turn is equivalent to AnnR�J� �⊆ AnnR�Hi� for every i by [23,Beispiel 1.1]. �
Example 3.8. It follows easily from Example 2.3 and Theorem 3.5 that over aright perfect local ring R, every finitely generated free R-module is almost injective.
Theorem 3.9. Let R be any ring. Then RR is almost injective if and only if everyfinitely generated free R-module is almost injective.
Proof. This follows immediately from Theorem 3.5. �
Proposition 3.10. (1) Let M be a module with Rad�M� = 0. Then M is almostinjective if and only if M is injective.
(2) Let M be a semisimple module. Then M is almost injective if and only if M isinjective.
(3) If R is a right V -ring, then every almost injective R-module M is injective.
Proof. (1) Suppose that M is almost injective. Then there is a submodule L of E�M�such that M is a supplement of L in E�M� and so E�M� = M + L and M ∩ L � M .Since Rad�M� = 0, we get M ∩ L = 0 and so E�M� = M ⊕ L. Hence M is injective.
(2) and (3) follow from (1) since in both cases Rad�M� = 0. �
A ring R is called a right Harada ring (right H-ring for short) if everyinjective right R-module is lifting. (See, for example, [5, 28.10] which gives somecharacterisations of right Harada rings.) Quasi-Frobenius rings and artinian serialrings are right H-rings. Note that every almost injective R-module is injective, overa right H-ring R.
Proposition 3.11. Let R be a ring such that every injective R-module is amplysupplemented. The following are equivalent:
(i) Every weakly injective R-module is injective;(ii) Every almost injective R-module is injective;(iii) R is a right H-ring.
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Proof. (i) ⇒ (ii) and (iii)⇒ (ii) are clear.
(ii) ⇒ (i) Let M be a weakly injective module. By Proposition 3.2, M iscoclosed in E�M�. Then, since E�M� is amply supplemented, M is a supplement inE�M� by [5, 20.3]. Hence M is almost injective by Proposition 3.2 and so injectiveby (ii).
(ii) ⇒ (iii) Let E be an injective module, and let M be a supplement in E.By hypothesis, M is injective. Hence M is a direct summand of E. Since E isamply supplemented, it follows that E is lifting by [5, 22.3]. Therefore, R is a rightH-ring. �
Corollary 3.12. The following are equivalent for a ring R:
(i) R is right perfect and every weakly injective R-module is injective;(ii) R is right perfect and every almost injective R-module is injective;(iii) R is a right H-ring.
Proof. This is a consequence of Propositions 2.1, 3.11, and [5, 27.24 and 28.10]. �
In particular, if R is a quasi-Frobenius ring or an artinian serial ring, thenevery almost injective R-module is injective.
Example 3.13. From Corollary 3.12, we see that if R is a right perfect ring whichis not a right H-ring, then there is an almost injective R-module which is notinjective. For an example of such a ring, consider the ring R = [ � �
0 �
]of 2× 2 upper
triangular real matrices with all diagonal entries rational. By [3, p. 318], R is rightperfect but not right artinian. It follows that R is not a right H-ring by [5, 28.10].
Example 3.14. Note that, in general, a ring whose almost injective modules areinjective need not be a right H-ring. In fact, if R is the ring of Example 2.10, thenR is a right V -ring but not right perfect and so not a right H-ring. However, everyalmost injective R-module is injective by Proposition 3.10.
Proposition 3.15. Let M = M1 + · · · +Mn be an irredundant sum of hollow modulesMi such that M is not an injective module. Then M is almost injective if and only ifthere is a submodule L of E�M� such that M1 + · · · +Mn + L = E�M� and this sum isirredundant.
Proof. Suppose that M is almost injective. Then there is a nonzero submodule L ofE�M� such thatM is a supplement of L in E�M�. By the minimality ofM , the sumM1 +· · · +Mn + L = E�M� is irredundant. Conversely, since the sumM1 + · · · +Mn + L =E�M� is irredundant and M1 is hollow, M1 is a supplement of M2 + · · · +Mn + L inE�M�. By the same token, we see that M2 is a supplement of M1 +M3 + · · · +Mn + Lin E�M�. Zöschinger [24, Lemma 1.3(a)] then shows that M1 +M2 is a supplement ofM3 + · · · +Mn + L in E�M�. By repeated application of this argument, we get that Mis a supplement of L in E�M�, as required. �
We denote the right annihilator of an element x in the ring R by annr �x�. Thusannr �x� = a ∈ R � xa = 0.
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Proposition 3.16. Let R be a local ring such that the module RR is almost injective.Then, for any x ∈ R, x is invertible if and only if annr �x� = 0.
Proof. If annr �x� = 0, then RR � xR and so, since RR is not small, x �∈ Rad�R�.Then R = xR = Rx, and so x is invertible. The converse is clear. �
Corollary 3.17. Let R be a right perfect local ring with maximal right ideal J . ThenJ = x ∈ R � annr �x� �= 0.
Proof. This follows from Example 2.3 and Proposition 3.16. �
Proposition 3.18. Let R be a commutative local ring such that the module RR isuniform. Let J be the maximal ideal of R. Then RR is almost injective if and only ifJ = x ∈ R � annr �x� �= 0.
Proof. The necessity follows Proposition 3.16.By Proposition 2.2, for the sufficiency we need only show that RR is not
a small module. Suppose not, so that R � E�RR�. Then R ≤ Rad�E�RR��. ButRad�E�RR�� = E�RR�J by [3, Corollary 15.18] and so R ≤ E�RR�J . Thus we mayexpress 1R, the identity of R, as 1R = ∑n
i=1 xi�i where �i ∈ J and xi ∈ E�RR�. Byhypothesis, annr ��i� �= 0 for each i and so, since RR is uniform,
⋂ni=1 annr ��i� �=
0. Let � ∈ ⋂ni=1 annr ��i� such that � �= 0. Then � = ∑n
i=1 xi�i� and so � = 0, acontradiction, as required. �
The ring detailed in [4] is an example of a non-self-injective chain ringsatisfying the conditions of Proposition 3.18.
4. F -ALMOST INJECTIVE MODULES
In [23], the author studied modules for which every factor module is weaklyinjective. He showed the following result.
Lemma 4.1. Every factor module of the module M is weakly injective if and only ifM is noncosingular.
A submodule N of the module M is called fully invariant if f�N� ≤ N for everyf ∈ EndR�M�.
The following example shows that a factor module of an almost injectivemodule, even by a fully invariant submodule, need not be almost injective in general.
Example 4.2. Note first that if R is a local ring with maximal right ideal J , thenthe R-module R/J is injective if and only if R is a division ring. In fact, if R/J is aninjective R-module, then every simple R-module is injective. This implies that R is aright V -ring and so J = 0. Therefore, the ring R is a division ring.
Now let R be the non-self-injective chain ring mentioned above, or let R be aright artinian local ring which is not quasi-Frobenius (see Example 2.7). Set S=R/J ,where J is the maximal right ideal of R. Then S is not injective and so is a smallmodule. It follows from Proposition 2.2 that S is not (weakly) almost injective. Notethat J is a fully invariant submodule of RR by [3, Proposition 9.14].
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We say that a module M is F -almost injective (FFI-almost injective) if the factormodule M/N is almost injective for every (fully invariant) submodule N ≤ M .
Example 4.3. Let R be a right hereditary ring and let M be an almost injective R-module. Since every factor module of E�M� is injective, it follows that M is F -almostinjective by Proposition 3.2 and [5, 20.5(1)].
Proposition 4.4. The following are equivalent for a ring R:
(i) Every R-module is F -almost injective;(ii) RR is F -almost injective;(iii) R is semisimple.
Proof. (ii) ⇒ (iii) Let M be a maximal right ideal of R. By hypothesis, R/M is analmost injective R-module. Since R/M is simple, it is injective by Proposition 3.10(2). Thus R is a right V -ring. Now let I be any right ideal of R. Since R/I is almostinjective and R is a right V -ring, R/I is injective by Proposition 3.10 (3). Then, by[18, Theorem, p. 649], the ring R is semisimple.
(iii) ⇒ (i) and (i) ⇒ (ii) are clear. �
Lemma 4.5. Let L be a small submodule of a module M . If M/L is an almost injectivemodule, then so is M .
Proof. If M/L is almost injective then it is a supplement in E�M�/L. Then there is asubmodule N ≤ E�M� such that L ≤ N and M/L is a supplement of N/L in E�M�/L.Since L � M , it is easy to check that M is a supplement of N in E�M�. �
Proposition 4.6. Let M = M1 +M2 such that M1 and M2 are F -almost injective.Then M is F -almost injective.
Proof. Let N be a submodule of M . Then �M1 + N�/N � M1/�N ∩M1� is almostinjective. Moreover, since M/N
�M1+N�/N� M/�M1 + N� = ��M1 + N�+M2�/�M1 + N� �
M2/M2 ∩ �M1 + N� is almost injective, taking into account Lemma 3.4, M/N isalmost injective. Therefore, M is F -almost injective. �
Proposition 4.7. Let M = M1 ⊕M2 where M1 and M2 are FFI-almost injectivemodules. Then M is FFI-almost injective.
Proof. Let U be a fully invariant submodule of M , and set Ui = U ∩Mi fori = 1� 2. It is easy to check that U = U1 ⊕ U2 and that Ui is a fully invariantsubmodule of Mi for both i. By assumption, each Mi/Ui is almost injective. SinceM/U � �M1/U1�⊕ �M2/U2�, it follows that M/U is almost injective by Theorem 3.5.Consequently, M is FFI-almost injective. �
We end this section with examples to show that the implications “injective ⇒F -almost injective” and “F -almost injective ⇒ injective” do not hold.
Example 4.8. Let R be a right perfect local ring which is not a division ring (seeExample 2.7). Then R has no nonzero F -almost injective modules. To see this,
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SUPPLEMENT SUBMODULES OF INJECTIVE MODULES 4399
suppose there exists a nonzero F -almost injective R-module M . Since R is rightperfect, we have Rad�M� �= M . Thus M contains a maximal submodule K. HenceM/K is simple almost injective and so is simple injective. By Example 4.2, R is adivision ring. This contradicts our assumption.
In particular, R has no nonzero injective R-modules which are F -almostinjective.
In order to construct an F -almost injective module which is not injective, weneed the following definitions.
As in [21], we say that a module M is almost finitely generated if M is notfinitely generated but every proper R-submodule of M is finitely generated.
A ring R is said to be an almost discrete valuation ring if R is a commutativelocal noetherian domain of Krull dimension 1 and the integral closure R of R inthe field of fractions Q�R� of R is a finitely generated R-module and is a discretevaluation ring.
Example 4.9. Consider the ring R = k��x2� x3��, where k is any field. It is easilyseen from [21, Example 5.1] that R is an almost discrete valuation ring which is nota discrete valuation ring.
Example 4.10. Let R be an almost discrete valuation ring with maximal idealJ and field of fractions Q�R� such that R is not a discrete valuation ring. It iswell known that the only indecomposable injective R-modules are E�R� � Q�R�and E�R/J�. By [8, Proposition 4], Q�R� is an almost finitely generated R-moduleand E�R/J� is hollow radical. By [8, Proposition 3], there is an ideal I of R suchthat E�R/J� � Q�R�/I . Since R is not a discrete valuation ring, R is not hereditary.By [22, 39.16], there is an R-submodule L of Q�R� such that the R-module M =Q�R�/L is not injective. Note that every factor module of Q�R� is divisible and soweakly injective by [23, Beispiel 2.13]. Since Rad�Q�R�� = Q�R� and IR is finitelygenerated, we have IR � Q�R� and hence the R-module Q�R� is hollow. Thus everyfactor module of Q�R� is hollow. Proposition 2.2 shows that every factor module ofQ�R� is almost injective. It follows that M is F -almost injective.
5. F -ALMOST INJECTIVE LIFTING MODULES
A module M is said to have the (strong) summand sum property, denoted brieflyby (SSSP) SSP, if the sum of (any family of) two direct summands of M is a directsummand of M .
Theorem 5.1. Let M be a lifting F -almost injective module. Then the following hold:
(1) If A and B are direct summands of M and f � A → B is any homomorphism, thenIm f is a direct summand of B;
(2) M has the SSSP.
Proof. (1) Let A and B be direct summands of M and f � A → B be anyhomomorphism. Clearly, A/Ker f � Im f and A/Ker f is isomorphic to a factormodule of M . Since M is F -almost injective, Im f is almost injective and so a
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4400 CLARK ET AL.
supplement submodule of B. Then, since B is lifting by [5, Lemma 22.6], Im f is adirect summand of B by [5, 22.3].
(2) This follows from (1), [1, Theorem 8] and [7, Proposition 4.9]. �
Let M be a module. Set S = EndR�M�. As in [11], the module M is called dualBaer if for every N ≤ M , there exists an idempotent e in S such that D�N� = eSwhere D�N� = � ∈ S � Im� ⊆ N.
Corollary 5.2. Every lifting F -almost injective module is dual Baer.
Proof. By Theorem 5.1 and [11, Theorem 2.1]. �
The following example shows that a dual Baer module need not be F -almostinjective in general.
Example 5.3. Let p be any prime integer. Consider the simple �-module M =�/p�. By [11, Corollary 3.5], M is dual Baer. On the other hand, M is not almostinjective since it is not injective.
A module M is called regular if every cyclic submodule of M is a directsummand of M .
Corollary 5.4.
(1) Let M be a lifting F -almost injective module. Then M is a direct sum of hollowmodules.
(2) If M is a regular lifting F -almost injective module, then M is semisimple.
Proof. This follows from Corollary 5.2 and [11, Corollaries 2.6(ii) and 2.7]. �
Lemma 5.5. Let I be any two-sided ideal of a ring R, and let M be an �R/I�-module.
(1) If MR is almost injective, then M�R/I� is almost injective.(2) If MR is F -almost injective, then M�R/I� is F -almost injective.
Proof. (1) Let E be any �R/I�-module such that M ≤ E. Clearly, E is an R-module and �R/I�-submodules of E are the same as the R-submodules of E. Byhypothesis, there is an R-submodule N of E such that MR is a supplement of NR
in ER. It is easily seen that M�R/I� is a supplement of N�R/I� in E�R/I�. It follows thatM�R/I� is almost injective.
(2) follows from (1) and the fact that �R/I�-submodules of M are the sameas the R-submodules of M . �
Theorem 5.6. Let L be a local module over a commutative ring R. Then the followingare equivalent:
(i) L is F -almost injective;(ii) L is simple injective;
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SUPPLEMENT SUBMODULES OF INJECTIVE MODULES 4401
(iii) L is simple almost injective;(iv) L is noncosingular (i.e., every factor module of L is weakly injective).
Proof. (i) ⇒ (ii) It is well known that L � R/I for some ideal I of R. SinceLR is F -almost injective, L is F -almost injective as an �R/I�-module by Lemma5.5. Therefore the module �R/I��R/I� is F -almost injective. Hence the ring R/I issemisimple by Proposition 4.4. Therefore, L is a simple R-module. Since L is asupplement in its injective hull, L is injective.
(ii) ⇒ (i) and (ii) ⇔ (iii) are clear.
(ii) ⇔ (iv) This follows from [12, Lemma 3.10]. �
Corollary 5.7. Let M be a lifting F -almost injective module over a commutative ringR. Then M = M1 ⊕M2 where Rad�M1� = M1 and M2 is semisimple injective.
Proof. From Corollary 5.4 it follows that M is a direct sum of hollow modules.Then M can be written as M = M1 ⊕M2 such that Rad�M1� = M1 and M2 =
⊕i∈I Li
is a direct sum of local modules Li. It follows from [5, Lemma 22.6] that the Li
are F -almost injective lifting. Proposition 3.10(2) and Theorem 5.6 now finish theproof. �
The following example shows that Theorem 5.6 and Corollary 5.7 do not holdif the commutativity of R is dropped.
Example 5.8. Let F be any field. Let R denote the ring of all upper triangular2× 2 matrices with entries in F . Consider the right R-module M = [
F F0 0
]. Then:
(1) The ring R is a (left and right) hereditary artinian serial ring and E�RR� =[F FF F
].
(See [6, Example 13.6] and [13, Example 3.43]).(2) The module M is cyclic artinian.(3) The module M is not semisimple since Soc�M� = [
0 F0 0
].
(4) The module M is projective since M ≤ RR and R is right hereditary.(5) The module M is lifting by [17, Theorem 4.41].(6) The module M is injective since it is a direct summand of E�RR�.(7) Every almost injective module is F -almost injective by Example 4.3. In
particular, M is F -almost injective.(8) The module M is a finite direct sum of local F -almost injective modules (see [5,
Corollary 22.12]).
Recall that a module M is called discrete if M is lifting and satisfies thefollowing condition:
�D2� For any submodule K of M with M/K isomorphic to a direct summandof M , K is a direct summand of M .
Proposition 5.9. Let M be a discrete F -almost injective module. Then the ringEndR�M� is von Neumann regular.
Proof. This follows from Lemma 4.1 and [12, Lemma 3.3]. �
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4402 CLARK ET AL.
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