Volume 2007, Article ID 10836,5pages doi:10.1155/2007/10836
Research Article
A Note on
⊕
-Cofinitely Supplemented Modules
Yongduo Wang and Qing Sun
Received 9 January 2007; Accepted 2 May 2007
Recommended by Francois Goichot
LetRbe a ring andMa rightR-module. In this note, we show that a quotient of an⊕ -cofinitely supplemented module is not in general⊕-cofinitely supplemented and prove that if a moduleMis an⊕-cofinitely supplemented multiplication module with Rad(M) M, then M can be written as an irredundant sum of local direct summand ofM. An extension of the result of Calisici and Pancar [1], here it is shown that an arbitrary module is cofinitely semiperfect if and only if it is an (amply) cofinitely supplemented by supplements which have projective covers.
Copyright © 2007 Y. Wang and Q. Sun. 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 and preliminaries
Throughout this paper,Ris an associative ring with identity and all modules are unital rightR-modules. We useN≤Mto indicate thatNis a submodule ofM.
M. InSection 2, we show that a quotient of an⊕-cofinitely supplemented module is not in general⊕-cofinitely supplemented. It is proven that if a moduleM is an⊕-cofinitely supplemented multiplication module with Rad(M)M, thenM can be written as an irredundant sum of local direct summand ofM. Calisici and Pancar [1] defined the con-cept of cofnitely semiperfect modules and proved that for a projective moduleM,M is cofinitely semiperfect if and only ifMis amply cofinitely supplemented. InSection 3, we will show that an arbitrary module is cofinitely semiperfect if and only if it is (amply) cofinitely supplemented by supplements which have projective covers. This extends [1, Theorem 2.3].
2.⊕-cofinitely supplemented modules
LetM be a module.M is called an⊕-cofinitely supplemented module if every cofinite submodule ofMhas a supplement that is a direct summand ofM.
We start with the following.
Example 2.1. LetRbe a commutative local ring which is not a valuation ring and let n≥2. By [3, Theorem 2], there exists a finitely presented indecomposable moduleM= R(n)/Kwhich cannot be generated by fewer thannelements. By [4, Corollary 1],R(n)is⊕
-cofinitely supplemented. However,Mis not⊕-cofinitely supplemented by [4, Proposition 2].
Example 2.2. LetRbe a commutative local ring which is not a valuation ring. Letaandb be elements ofR, neither of them divides the other. By taking a suitable quotient ring, we may assume that (a)∩(b)=0 andam=bm=0, wheremis the maximal ideal ofR. Let Fbe a free module with generatorsx1,x2, andx3. LetK be the submodule generated by
ax1−bx2andM=F/K. Thus,M=(Rx1⊕Rx2⊕Rx3)/R(ax1−bx2)=(Rx1+Rx2)⊕Rx3.
Following by [5, Example 2.3],Rx1⊕Rx2⊕Rx3is⊕-cofinitely supplemented, butM is
not⊕-cofinitely supplemented.
The above two examples show that a factor module of an⊕-cofinitely supplemented module is not in general⊕-cofinitely supplemented.
To deal with a special case of factor modules of⊕-cofinitely supplemented modules, we need the following lemma.
Lemma 2.3 [5, Lemma 2.4]. LetMbe a module andUa fully invariant submodule ofM. IfM=M1⊕M2, thenU=(U∩M1)⊕(U∩M2).
Proposition 2.4. LetMbe a module andUa fully invariant submodule ofM. IfMis an ⊕-cofinitely supplemented module, thenM/U is an⊕-cofinitely supplemented module. If, moreover,Uis a cofinite direct summand ofM, thenUis also an⊕-cofinitely supplemented module.
U=(U∩N)⊕(U∩N). Thus we have (N+U)∩(N +U)=U and ((N+U)/U)⊕ ((N +U)/U)=M/U, and hence (N+U)/Uis a direct summand ofM/U. SoM/Uis an ⊕-cofinitely supplemented module.
Now suppose thatUis a cofinite direct summand ofM. Then there is a submodule U such thatM=U⊕U andU is finitely generated. Let V a cofinite submodule of U. Note thatM/V=(U⊕U)/V∼=U/V⊕U is finitely generated so thatV is a cofinite submodule ofM. SinceMis an⊕-cofinitely supplemented module, there are submodules K and K of M such that M=K⊕K ,M=V+K, and V∩K K. Thus U=V+ (U∩K). But U=(U∩K)⊕(U∩K ), and hence U∩K is a direct summand of U. Moreover,V∩(U∩K)=V∩KK. ThenV∩(U∩K)(U∩K). Therefore,U∩K is a supplement ofV inU and it is a direct summand ofU. Thus, U is an⊕-cofinitely
supplemented module.
Remark 2.5. LetM be an⊕-cofinitely supplemented module. It is clear thatM/Rad(M) andM/Soc(M) are also⊕-cofinitely supplemented modules.
Lemma 2.6. LetMbe an⊕-cofinitely supplemented module. IfMcontains a maximal sub-module, thenMcontains a local direct summand.
Proof. LetLbe a maximal submodule ofM. SinceM is an⊕-cofinitely supplemented module, there exists a direct summandK ofMsuch thatK is a supplement ofLinM. Then for any proper submoduleXofK,Xis contained inLsinceLis a maximal sub-module andL+Xis a proper submodule ofMby minimality ofK. HenceX≤L∩Kand XK. ThusKis a hollow module, and the lemma is proven.
Theorem 2.7. Let M be an ⊕-cofinitely supplemented multiplication module with Rad(M)M, then M can be written as an irredundant sum of local direct summands ofM.
Proof. Since Rad(M)M,Mcontains a maximal submodule, and henceMcontains a local direct summand. Let N be the sum of all local direct summands ofM. IfN is a proper submodule ofM, then there is a maximal submoduleLofM such thatN≤Lby [6, Theorem 2.5]. LetPbe a direct summand ofMsuch thatPis a supplement ofLinM. Note thatPis a local module, and hence it is contained inN, soM=L+P≤L+N=L. This is a contradiction. Hence we haveN=M. Now letM=Σi∈ILi, where eachLiis a
local direct summand ofM. Then M
Rad(M)=Σi∈I L
i+ Rad(M)
Rad(M)
(2.1)
and each
Li+ Rad(M) Rad(M) ∼=
Li
Li∩Rad(M) (2.2)
is simple. Hence
M Rad(M)=
k∈K
L
k+ Rad(M) Rad(M)
for some subsetK⊆I. ThusM=Σk∈KLk since Rad(M)M. It is clear that the sum Σk∈KLkis irredundant.
Remark 2.8. The moduleM=(Rx1+Rx2)⊕Rx3in Example 2.2is not an⊕-cofinitely
supplemented module. On the other hand, M can be written as follows: M=(Rx1+
Rx2)⊕R(x1−x3);M=(Rx1+Rx2)⊕R(x2−x3); andM=R(x1−x3) +R(x2−x3) +Rx3.
Therefore,Mis an irredundant sum of local direct summands ofM.
3. Cofinitely semiperfect modules
LetMbe a module.Mis called semiperfect if every factor module ofMhas a projective cover. Calisici and Pancar [1] introduced the notion of cofinitely semiperfect modules as a proper generalization of semiperfect modules.Mis called a cofinitely semiperfect module if every finitely generated factor module ofMhas a projective cover. The following result generalizes [1, Theorem 2.3].
Theorem 3.1. The following statements are equivalent for a moduleM: (1)Mis a cofinitely semiperfect module;
(2)Mis amply cofinitely supplemented by supplements which have projective covers; (3)Mis cofinitely supplemented by supplements which have projective covers.
Proof. “(1)⇒(2)” LetM=N+LwithN cofinite. LetP be a projective cover ofM/N with epimorphism f. SinceP is projective andM/N is isomorphic toL/D, whereD is the intersection ofNandL, the epimorphism f lifts to a homomorphismgfromPtoL. Next, we show that Imgis a supplement ofNinMandPis a projective cover of Img. Since Kerf Pandg(Kerf)=Img∩D, Img∩DImg. Since f is an epimorphism, Img+ D=L. Thus, Imgis a supplement ofDinL. It is easy to verify that Img is a supplement ofNinM. Since Kerg≤Kerf, KergPand hencePis a projective cover of Imgwhich is clearly contained inL.
“(2)⇒(3)” is clear.
“(3)⇒(1)” LetNbe a cofinite submodule ofMandLis a supplement ofNinM. Since ifDdenotes the intersection ofNandL,Lis a cover ofL/D, then any projective cover of Lis a projective cover ofL/D, which is isomorphic toM/N. This completes the proof.
Acknowledgment
This research was supported by a Doctoral Initial Fund of Lanzhou University of Tech-nology.
References
[1] H. Calisici and A. Pancar, “Cofinitely semiperfect modules,” Siberian Mathematical Journal, vol. 46, no. 2, pp. 359–363, 2005.
[2] H. Calisici and A. Pancar, “⊕-cofinitely supplemented modules,” Czechoslovak Mathematical
Journal, vol. 54(129), no. 4, pp. 1083–1088, 2004.
[3] R. B. Warfield Jr., “Decomposability of finitely presented modules,” Proceedings of the American
[4] A. Idelhadj and R. Tribak, “A dual notion of CS-modules generalization,” in Algebra and Number
Theory, vol. 208 of Lecture Notes in Pure and Applied Mathematics, pp. 149–155, Marcel Dekker,
New York, NY, USA, 2000.
[5] A. Idelhadj and R. Tribak, “On some properties of ⊕-supplemented modules,” International
Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 69, pp. 4373–4387, 2003.
[6] Z. A. El-Bast and P. F. Smith, “Multiplication modules,” Communications in Algebra, vol. 16, no. 4, pp. 755–779, 1988.
Yongduo Wang: Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
Email address:[email protected]
Qing Sun: Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
