AGQP-Injective Modules

Volume 2008, Article ID 469725,7pages doi:10.1155/2008/469725

Research Article

AGQP-Injective Modules

Zhanmin Zhu¹ and Xiaoxiang Zhang²

1Department of Mathematics, Jiaxing University, Jiaxing, Zhejiang 314001, China

2Department of Mathematics, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Zhanmin Zhu,zhanmin [email protected] Received 23 December 2007; Revised 20 April 2008; Accepted 20 June 2008

Recommended by Robert Lowen

LetRbe a ring and letMbe a rightR-module withSEndMR.Mis called almost general quasi- principally injectiveor AGQP-injective for shortif, for any 0/s∈S, there exist a positive integern and a left idealXsⁿofSsuch thatsⁿ/0 andlSKersⁿ Ssⁿ⊕Xsⁿ. Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additional conditions are studied.

Copyrightq2008 Z. Zhu and X. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Throughout R is an associative ring with identity, and all modules are unitary. Recall that a ring R is called right principally injective 1 or right P-injective for short if, every homomorphism from a principal right ideal ofRtoRcan be extended to an endomorphism ofR, or equivalently,lra Rafor alla∈R. The concept of right P-injective rings has been generalized by many authors. For example, in2,3, right P-injective rings are generalized in two directions, respectively. Following2, a ringRis called right GP-injective if, for any 0/a ∈ R, there exists a positive integernsuch thataⁿ/0 and any rightR-homomorphism fromaⁿR toR can be extended to an endomorphism ofR. Note that GP-injective rings are also called YJ-injective in 4. From 5, we know that GP-injective rings need not to be P- injective. Following3, a rightR-moduleMR withS EndMRis called quasiprincipally injective or QP-injective for short if, every homomorphism from anM-cyclic submodule of Mto M can be extended to an endomorphism ofM, or equivalently, lSKers Ss for alls ∈ S. In 1998, Page and Zhou 6 generalized the concept of GP-injective rings to that of AGP-injective rings. According to 6, a ring R is called right AGP-injective if, for any 0/a ∈ R, there exist a positive integer n and a left ideal X_aⁿ such that aⁿ/0 and lraⁿ Raⁿ⊕Xaⁿ. In7, the first author introduced the notion of GQP-injective modules which can be regarded as the generalization of GP-injective rings and QP-injective modules.

According to7, a rightR-moduleMwithS EndMRis called GQP-injective if, for any

0/s ∈ S, there exists a positive integer nsuch thatsⁿ/0 and any rightR-homomorphism from sⁿM to M can be extended to an endomorphism of M, or equivalently, for any 0/s ∈ S, there exists a positive integer nsuch thatsⁿ/0 andlSKersⁿ Ssⁿ. The nice structure of AGP-injective rings and GQP-injective modules draws our attention to define almost GQP-injective modules, in a similar way to AGP-injective rings, and to investigate their properties.

2. Results

Definition 2.1. LetM_Rbe a rightR-module withSEndMR. Then,Mis said to be almost general quasiprincipally injectivebriefly, AGQP-injectiveif, for any 0/s∈S, there exist a positive integernand a left idealXsⁿ ofSsuch thatsⁿ/0 andlSKersⁿ Ssⁿ⊕Xsⁿ.

Clearly, a ring R is right AGP-injective if and only if R_R is AGQP-injective, GQP- injective modules are AGQP-injective.

Our next result gives the relationship between the AGQP-injectivity of a module and the AGP-injectivity of its endomorphism ring.

Theorem 2.2. LetMRbe a rightR-module withSEndMR. Then, 1ifSis right AGP-injective, thenMRis AGQP-injective;

2ifMR is AGQP-injective andMgenerates Kersfor eachs ∈ S, thenSis right AGP- injective.

Proof. 1Suppose thatSis right AGP-injective then for any 0/s ∈ S, there exist a positive integernand a left idealIsⁿ ofSsuch thatsⁿ/0 andlSrSsⁿ Ssⁿ⊕Isⁿ. Ifa∈ lSKersⁿ andb∈rSsⁿ, thensⁿb0, that is,bM⊆Kersⁿ. Hence,abM0, that is,ab0. This shows thatlSKersⁿ ⊆ lSrSsⁿ. Therefore, we haveSsⁿ ⊆lSKersⁿ⊆ Ssⁿ⊕I_sⁿ, which guarantees that

lSKersⁿ Ssⁿ⊕lSKersⁿ∩I_sⁿ. 2.1

Thus,1is proved.

2Suppose thatM_R is AGQP-injective then for any 0/s ∈ S, there exist a positive integernand a left idealX_sⁿ ofSsuch thatsⁿ/0 andlSKersⁿ Ssⁿ⊕X_sⁿ. Assume that a∈lSrSsⁿand Kersⁿ

t∈TtMfor some subsetT ofS. It is easy to see thatat0 for eacht∈T, so we haveax 0 for eachx∈Kersⁿ. This implies thatlSrSsⁿ⊆lSKersⁿ, from which we have

Ssⁿ⊆lSrSsⁿ⊆lSKersⁿ Ssⁿ⊕X_sⁿ, 2.2

and hence

lSrSsⁿ Ssⁿ⊕lSrSsⁿ∩Xsⁿ. 2.3 Therefore,Sis right AGP-injective.

Recall that a moduleNis calledM-cyclic3, if it is a homomorphic image ofM. Let SEndMR, following8, we writeWS {s∈S |Kers⊆^essM}.

Theorem 2.3. LetMRbe an AGQP-injective module withSEndMR. Then, 1WS⊆JS,

2if every nonzero submodule ofMcontains a nonzeroM-cyclic submodule, thenWS JS.

Proof. 1Lets∈WS. Then, for eacht∈S,ts∈WSand so 1−ts /0. SinceMRis AGQP- injective, there exist a positive integernand a left ideal X_1−tsⁿ such that1−tsⁿ/0 and lSKer1−tsⁿ S1−tsⁿ⊕X_1−tsⁿ. Note that1−tsⁿ 1−ufor someu ∈WS. Since Keru∩Ker1−u 0, we have Ker1−u 0, and thenSS1−u⊕X_1−u. So 1exfor somee∈S1−uandx∈X_1−u, it follows thate²eandS1−u Se⊕S1−e∩S1−u Se.

Therefore, 1−uvefor somev∈S, since Keruis essential inM_R, ife /1, then there exists a nonzero element1−em∈1−eM∩Keru, and hence1−u1−em 1−em. But 1−u1−emve1−em0, a contradiction. Soe1,and hence 1−uis left invertible, which impliess∈JS.

2We need only to prove that JS ⊆ WS. Let s ∈ JS. Ifs /∈WS, then there exists 0/t∈Ssuch that Kers∩tM 0 by hypothesis. Clearly,st /0 and Kerst Kert.

SinceM_R is AGQP-injective, there exist a positive integernand a left idealX_stⁿ such that stⁿ/0 and

lSKerstⁿ Sstⁿ⊕X_stⁿ. 2.4

If m ∈ Kerstⁿ, then stⁿ⁻¹m ∈ Kerst Kert, and som ∈ Kertstⁿ⁻¹. This shows that Kerstⁿ Kertstⁿ⁻¹. Hence, tstⁿ⁻¹ ∈ Sstⁿ⊕X_stⁿ. Write tstⁿ⁻¹ ustⁿv, whereu ∈ S, v ∈ X_stⁿ. Then1−uststⁿ⁻¹ v, which gives thatstⁿ s1−us⁻¹v ∈ Sstⁿ∩X_stⁿ 0, a contradiction.

Corollary 2.4see6, Corollary 2.3. IfRis a right AGP-injective ring, thenJR ZRR. Following9, for a setX⊆HomNR, MR, the submodule

KerX∩{Kerg | g ∈X} 2.5

ofN is called anM-annihilator submodule ofN. By7, Lemma 9andTheorem 2.3, we have the following corollary.

Corollary 2.5. Let MR be an AGQP-injective module with S EndMR. If every nonzero submodule ofM contains a nonzero M-cyclic submodule, andM/SocM satisfies ACC on M- annihilator submodules, thenJSis nilpotent.

Recall that a moduleMR is said to be a GC2 module10if every submoduleN ≤ M withN ∼Mis a direct summand ofM. For convenience, we writeN |Mto denote thatN is a direct summand ofM.

Theorem 2.6. LetMRbe an AGQP-injective module. Then,

1if M1 and M2 are submodules ofM such that M1 ⊆ M2 and M1 ∼ M2 | M, then M₁|M. In particularMis a GC2 module;

2ifM1andM2are simple submodules ofMsuch thatM1∼M2|M, thenM1 |M.

Proof. 1LetSEndMR. It is trivial in caseM₁ 0. Now suppose thatM₁/0 andM₂ ∼^f M₁. ThenM₁aMandM₂ eM, wheree²e∈Sandafe. SinceMRis AGQP-injective, there exist a positive integernand a left idealXaⁿsuch thataⁿ/0 andlSKeraⁿ Saⁿ⊕Xaⁿ. Leta⁰ e, thenf⁻¹aⁱ¹M aⁱMi0,1, . . . , n−1sinceM₁⊆M₂eM. So we have

aⁱM|aⁱ⁻¹M⇐⇒f⁻¹aⁱ¹M|f⁻¹aⁱM⇐⇒aⁱ¹M|aⁱM i1, . . . , n−1. 2.6

Consequently,aM | eM ⇔ a²M | aM ⇔ · · · ⇔ aⁿM | aⁿ⁻¹M. Thus, to showaM | M, it suffices to show thataⁿM | M. Note thata|eM : eM → eM is monic andaⁿm aⁿem for everym ∈M,eM ^a

∼n

aⁿMand hence Keraⁿ Kere. It follows thate ∈lSKere lSKeraⁿ Saⁿ ⊕Xaⁿ. Now, lete baⁿxwith b ∈ S andx ∈ Xaⁿ, then aⁿ aⁿe aⁿbaⁿaⁿxaⁿbaⁿ. Finally, letgaⁿb, theng² gandaⁿMgMas required.

2Let M₂ e₁M, wheree²₁ e₁ ∈ S, and letM₂ ^f∼¹ M₁. ThenM₁ a₁M, where a₁ f₁e₁. SinceM_Ris AGQP-injective, there exist a positive integern₁and a left idealX_aⁿ1 1

such thataⁿ₁¹/0 andlSKeraⁿ₁¹ Saⁿ₁¹⊕X_aⁿ¹

1 . Note that 0/aⁿ₁¹M⊆a₁M,anda₁Mis simple.

We haveaⁿ₁¹Ma1M. Clearly, Kere1 Kera1becausef1is a monomorphism. Sincea1M is simple, Kera1is a maximal submodule ofM. But Kera1⊆ Keraⁿ₁¹/M, so Kera1

Keraⁿ₁¹and then Kere1 Keraⁿ₁¹. It follows thate₁∈lSKere1 lSKeraⁿ₁¹ Saⁿ₁¹⊕ X_aⁿ1

1 . Now, lete1 b1aⁿ₁¹ywithb1 ∈Sandy∈X_aⁿ1

1 , thenaⁿ₁¹ aⁿ₁¹e1 aⁿ₁¹b1aⁿ₁¹aⁿ₁¹y aⁿ₁¹b1aⁿ₁¹. Finally, letg1aⁿ₁¹b1, theng₁² g1andM1 a1Maⁿ₁¹Mg1Mas required.

Recall that a moduleMis said to be weakly injective11if, for any finitely generated submoduleN≤EM, there existsX≤EMsuch thatN⊆X∼M.

Corollary 2.7. LetMbe a finitely generated module. Then,Mis injective if and only ifMis weakly injective and AGQP-injective. In particular, a ringRis right self-injective if and only ifR_Ris weakly injective and AGP-injective.

Proof. We need only to prove the sufficiency. Letx∈EM. Then, there existsX⊆EMsuch thatMxR⊆X∼M. Hence,Xis AGQP-injective andM|Xfollows fromTheorem 2.61.

ButMis essential inEM, soMXand hencex∈M.

Corollary 2.8. LetMRbe an AGQP-injective module withSEndMR. 1IfMRis of finite Goldie dimension, then S is semilocal.

2IfM_Ris a noetherian self-generator, thenSis semiprimary.

Proof. 1SinceMR is AGQP-injective, it satisfies the GC2-condition byTheorem 2.61and then1follows immediately by12, Lemma 1.1.

2By1andCorollary 2.5.

Recall that ifMandUare two rightR-modules, thenUis called M-projective in case for each epimorphismg : M_R → N_R and each homomorphism γ : U_R → N_R, there is an R-homomorphismγ :UR →MRsuch thatγ gγ. A moduleMRis called quasiprojective if it isM-projective.

LetRbe a ring. Recall that an elementa∈Ris calledπ-regular if there exists a positive integermsuch thata^m a^mba^m 13for someb∈R. An elementx∈Ris called generalized π-regular if there exists a positive integer n such thatxⁿ xⁿyx for some y ∈ R. A ring R is calledπ-regularresp., generalizedπ-regularif every element in Risπ-regularresp., generalizedπ-regular. IfAis a subset ofR, then we say thatAis regular if every element in Ais regular.

Proposition 2.9. LetM_R be quasiprojective withS EndMR. Then,Sis regular if and only if M_Ris AGQP-injective andsMisM-projective for everys∈S.

Proof. Assume that S is regular. Then, every right ideal of S is a direct summand of SS, and so every homomorphism from a principal right ideal of S to S can be extended to an endomorphism of S. Hence, S is right P-injective and then right AGP-injective. By Theorem 2.2,MR is AGQP-injective. The regularity of Salso implies that sMis a direct summand ofMby14, Theorem 37.7. ButMis quasiprojective, sosMisM-projective for everys∈S.

Conversely, supposeMRis AGQP-injective andsMisM-projective for everys∈S.

Then for any 0/a∈S, by the AGQP-injectivity ofM_R, there exist a positive integernand a left idealX_aⁿ ofSsuch thataⁿ/0 andlSKeraⁿ Saⁿ⊕X_aⁿ. SinceaⁿMisM-projective, Keraⁿ eMfor somee²e∈S. Then, we haveS1−e lSeM lSKeraⁿ Saⁿ⊕Xaⁿ, and so 1−ebaⁿxfor someb∈Sandx∈X_aⁿ. Thus,aⁿaⁿ1−e aⁿbaⁿaⁿxaⁿbaⁿ. This proves thatSisπ-regular and hence generalizedπ-regular. Clearly,N₁S {0/a ∈ S | a² 0}is regularin this case,nmust be equal to 1. Therefore or,Sis regular by13, Theorem 2.2.

Recall that a moduleMRis called an IN-module15iflSA∩B lSA lSBfor any submodulesAandBofM, whereSEndMR.

Proposition 2.10. LetM_Rbe an AGQP-injective IN-module withSEndMR. Then,Sis regular if and only ifWS 0.

Proof. By Theorem 2.3, we need only to prove the sufficiency. Let 0/a ∈ S. Since M_R is AGQP-injective, there exist a positive integernand a left idealX_aⁿ ofSsuch thataⁿ/0 and lSKeraⁿ Saⁿ⊕Xaⁿ. SinceWS 0, Keraⁿis not essential inMand then there exists a nonzero submoduleKsuch that Keraⁿ⊕Kis essential inM. Moveover, we also have

lSKeraⁿ lSK lSKeraⁿ∩K S,

lSKeraⁿ∩lSK⊆lSKeraⁿ K 0, 2.7

becauseMRis an IN-module andWS 0. Thus,

SlSKeraⁿ⊕lSK Saⁿ⊕Xaⁿ⊕lSK. 2.8

Let 1baⁿxwithb∈S,x∈Xaⁿ⊕lSK, thenaⁿaⁿbaⁿ. It follows thatSis regular by the last part of the proof ofProposition 2.9.

Lemma 2.11. LetMRbe an AGQP-injective module in which every nonzero submodule contains a nonzeroM-cyclic submodule andSEndMR. Ifs /∈WS, then the inclusion Kers⊆Kers− stsis strict for somet∈S.

Proof. Ifs /∈WS, then Kers∩K0 for some nonzero submoduleKofM, and so Kers∩ sM 0 for some 0/s ∈ Sby hypothesis. Clearly, ss/0. Since MR is AGQP-injective, there exist a positive integernand a left idealX_ssⁿ such thatssⁿ/0 andlSKerssⁿ Sssⁿ⊕X_ssⁿ. Thus,

sssⁿ⁻¹∈lSKersssⁿ⁻¹ lSKerssⁿ Sssⁿ⊕X_ssⁿ. 2.9

Writesssⁿ⁻¹ tssⁿx,wheret∈Sandx∈X_ssⁿ, then1−tssssⁿ⁻¹xand hence 1−stssⁿ s−stssssⁿ⁻¹sx∈Sssⁿ∩X_ssⁿ. 2.10 This means thats−stssssⁿ⁻¹ 0. It is obvious that Kers ⊆ Kers−sts. Note that sssⁿ⁻¹Mis contained in Kers−stsbut not contained in Kers, the inclusion Kers⊆ Kers−stsis strict.

Theorem 2.12. LetMR be AGQP-injective withSEndMR. If every nonzero submodule ofM contains a nonzeroM-cyclic submodule, then the following conditions are equivalent:

1Sis right perfect;

2for any sequence{s1, s₂, . . .} ⊆S, the chain Kers1⊆Kers2s₁⊆ · · · terminates.

Proof. ByTheorem 2.3,Lemma 2.11, and 16, Lemma 2.8, one can complete the proof in a similar way to that of16, Theorem 2.9.

Acknowledgment

The authors are very grateful to the referees for their useful comments and suggestions.

References

1 W. K. Nicholson and M. F. Yousif, “Principally injective rings,” Journal of Algebra, vol. 174, no. 1, pp.

77–93, 1995.

2 S. B. Nam, N. K. Kim, and J. Y. Kim, “On simple GP-injective modules,” Communications in Algebra, vol. 23, no. 14, pp. 5437–5444, 1995.

3 N. V. Sanh, K. P. Shum, S. Dhompongsa, and S. Wongwai, “On quasi-principally injective modules,”

Algebra Colloquium, vol. 6, no. 3, pp. 269–276, 1999.

4 R. Yue Chi Ming, “On injectivity andp-injectivity,” Journal of Mathematics of Kyoto University, vol. 27, no. 3, pp. 439–452, 1987.

5 J. Chen, Y. Zhou, and Z. Zhu, “GP-injective rings need not be P-injective,” Communications in Algebra, vol. 33, no. 7, pp. 2395–2402, 2005.

6 S. S. Page and Y. Zhou, “Generalizations of principally injective rings,” Journal of Algebra, vol. 206, no.

2, pp. 706–721, 1998.

7 Z. Zhu, “On general quasi-principally injective modules,” Southeast Asian Bulletin of Mathematics, vol.

30, no. 2, pp. 391–397, 2006.

8 W. K. Nicholson, J. K. Park, and M. F. Yousif, “Principally quasi-injective modules,” Communications in Algebra, vol. 27, no. 4, pp. 1683–1693, 1999.

9 N. V. Dung, D. V. Huynh, P. F. Smith, and R. Wisbauer, Extending Modules, vol. 313 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994.

10 M. F. Yousif and Y. Zhou, “Rings for which certain elements have the principal extension property,”

Algebra Colloquium, vol. 10, no. 4, pp. 501–512, 2003.

11 S. K. Jain and S. R. L ´opez-Permouth, “Rings whose cyclics are essentially embeddable in projective modules,” Journal of Algebra, vol. 128, no. 1, pp. 257–269, 1990.

12 Y. Zhou, “Rings in which certain right ideals are direct summands of annihilators,” Journal of the Australian Mathematical Society, vol. 73, no. 3, pp. 335–346, 2002.

13 J. Chen and N. Ding, “On regularity of rings,” Algebra Colloquium, vol. 8, no. 3, pp. 267–274, 2001.

14 R. Wisbauer, Foundations of Module and Ring Theory, vol. 3 of Algebra, Logic and Applications, Gordon and Breach Science, Philadelphia, Pa, USA, German edition, 1991.

15 R. Wisbauer, M. F. Yousif, and Y. Zhou, “Ikeda-Nakayama modules,” Contributions to Algebra and Geometry, vol. 43, no. 1, pp. 111–119, 2002.

16 Z. Zhu, Z. Xia, and Z. Tan, “Generalizations of principally quasi-injective modules and quasiprinci- pally injective modules,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no.

12, pp. 1853–1860, 2005.