Generalized ⊕-supplemented modules

Let R be a ring and M be a left R-module. M is called generalized ⊕- supplemented if every submodule of M has a generalized supplement that is a direct summand of M. In this paper we give various properties of such modules. We show that any finite direct sum of generalized ⊕-supplemented modules is...

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Date:	2010
Main Authors:	Calısıcı, H., Turkmen, E.
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Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2010
Series:	Algebra and Discrete Mathematics
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Cite this:	Generalized ⊕-supplemented modules / H. Calısıcı, E. Turkmen // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 10–18. — Бібліогр.: 10 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1548342025-02-09T15:40:44Z Generalized ⊕-supplemented modules Calısıcı, H. Turkmen, E. Let R be a ring and M be a left R-module. M is called generalized ⊕- supplemented if every submodule of M has a generalized supplement that is a direct summand of M. In this paper we give various properties of such modules. We show that any finite direct sum of generalized ⊕-supplemented modules is generalized ⊕-supplemented. If M is a generalized ⊕-supplemented module with (D3), then every direct summand of M is generalized ⊕-supplemented. We also give some properties of generalized cover. 2010 Article Generalized ⊕-supplemented modules / H. Calısıcı, E. Turkmen // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 10–18. — Бібліогр.: 10 назв. — англ. 2000 Mathematics Subject Classification:16D10,16D99. https://nasplib.isofts.kiev.ua/handle/123456789/154834 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Let R be a ring and M be a left R-module. M is called generalized ⊕- supplemented if every submodule of M has a generalized supplement that is a direct summand of M. In this paper we give various properties of such modules. We show that any finite direct sum of generalized ⊕-supplemented modules is generalized ⊕-supplemented. If M is a generalized ⊕-supplemented module with (D3), then every direct summand of M is generalized ⊕-supplemented. We also give some properties of generalized cover.
format	Article
author	Calısıcı, H. Turkmen, E.
spellingShingle	Calısıcı, H. Turkmen, E. Generalized ⊕-supplemented modules Algebra and Discrete Mathematics
author_facet	Calısıcı, H. Turkmen, E.
author_sort	Calısıcı, H.
title	Generalized ⊕-supplemented modules
title_short	Generalized ⊕-supplemented modules
title_full	Generalized ⊕-supplemented modules
title_fullStr	Generalized ⊕-supplemented modules
title_full_unstemmed	Generalized ⊕-supplemented modules
title_sort	generalized ⊕-supplemented modules
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	https://nasplib.isofts.kiev.ua/handle/123456789/154834
citation_txt	Generalized ⊕-supplemented modules / H. Calısıcı, E. Turkmen // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 10–18. — Бібліогр.: 10 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT calısıcıh generalizedsupplementedmodules AT turkmene generalizedsupplementedmodules
first_indexed	2025-11-27T13:08:35Z
last_indexed	2025-11-27T13:08:35Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 2. pp. 10 – 18 c© Journal “Algebra and Discrete Mathematics” Generalized ⊕-supplemented modules Hamza Çalışıcı and Ergül Türkmen Communicated by V. V. Kirichenko Abstract. Let R be a ring and M be a left R-module. M is called generalized ⊕- supplemented if every submodule of M has a generalized supplement that is a direct summand of M . In this paper we give various properties of such modules. We show that any finite direct sum of generalized ⊕-supplemented modules is generalized ⊕-supplemented. If M is a generalized ⊕-supplemented module with (D3), then every direct summand of M is generalized ⊕-supplemented. We also give some properties of generalized cover. 1. Introduction In this note R will be an associative ring with identity and all modules unital left R-modules. Let M be an R-module. The notation N ≤ M means that N is a submodule of M. Rad (M) will indicate Jacobson radical of M. A submodule N of an R-module M is called small in M (notation N << M), if N + L 6= M for every proper submodule L of M . An epimorphism f : K → M is called a small cover (cover in [9]) if Ker f << K. Let M be an R-module and let N and K be any submodules of M . K is called a supplement of N in M if K is minimal with respect to M = N + K. K is a supplement of N in M iff M = N + K and N ⋂ K << K(see [8]). Following [8], M is called supplemented if every submodule of M has a supplement in M , and is called amply supplemented (supplemented in [6]) if for any two submodules U and V of M with M = U + V , V contains a supplement of U in M . Cleary amply supplemented modules are supplemented. If M = N +K 2000 Mathematics Subject Classification: 16D10,16D99. Key words and phrases: generalized cover, generalized supplemented module, ⊕-supplemented module, generalized ⊕-supplemented module. H. Çalı ş ıcı , E. Türkmen 11 and N ⋂ K << M , then K is called a weak supplement of N in M(see [5]). Then clearly N is a weak supplement of K, too. A module M is called weakly supplemented if every submodule of M has a weak supplement in M . Let M be an R-module and let N and K be any submodules of M with M = N +K. If (N ⋂ K ⊆ Rad (M))N ⋂ K ⊆ Rad (K) then K is called a (weak) generalized supplement of N in M . Since Rad (K) is the sum of all small submodules of K, every supplement submodule is a generalized supplement in M . Following [9], M is called generalized supplemented or briefly GS- module if every submodule N of M has a generalized supple- ment K in M , and it is called generalized amply supplemented or briefly GAS-module in case M = K +L implies that K has a generalized supple- ment L′ ≤ L. Clearly every (amply) supplemented module is generalized (amply) supplemented. In [7], a module M is called weakly generalized supplemented or briefly WGS-module if every submodule K of M has a weak generalized supplement N in M . For characterizations of generalized (amply) supplemented and weakly generalized supplemented modules we refer to [7] and [9]. Recall from [1] that an epimorphism f : P → M is called a generalized cover if Ker f ⊆ Rad (P ), and a generalized cover f : P → M is called generalized projective cover in case P is a projective module. Clearly every small cover is a generalized cover. In [1], M is called (generalized) semiperfect if every factor module of M has a (generalized) projective cover. The concepts of (generalized) semiperfect modules were introduced in [1] and [9]. This note consists of two sections. We obtain some properties of generalized cover in section 2. In section 3 we introduce generalized ⊕- supplemented modules. We show that every finite direct sum of generalized ⊕-supplemented modules is generalized ⊕-supplemented. 2. Generalized cover It was shown in [9, Lemma 1.1] that if f : M → N and g : N → K are generalized covers, then gf : M → K is a generalized cover, too. We prove that the converse of this fact is also true. Proposition 2.1. If f : M → N and g : N → K are two epimorphisms, then f and g are generalized covers if and only if gf : M → K is a generalized cover. Proof. (⇒) Let m ∈ Ker gf . Then (gf) (m) = 0 and f (m) ∈ Ker g ⊆ Rad (N). Note that Rf (m) << N . Suppose that m /∈ Rad (M). Then there exists a maximal submodule P of M such that P +Rm = M . Then 12 Generalized ⊕-supplemented modules f (P ) +Rf (m) = N , and since Rf (m) << N it follows that f (P ) = N . Hence P = f−1 (f (P )) = P +Ker f = M . This is a contraction. (⇐) Let m ∈ Ker f . Then g (f (m)) = 0 and by assumption, m ∈ Ker gf ⊆ Rad (M), i.e. Ker f ⊆ Rad (M). Let n ∈ Ker g. Since f is an epimorphism there exists an element m of M such that f (m) = n. Then (gf) (m) = g (n) = 0 and hence m ∈ Ker gf ⊆ Rad (M), which implies n = f (m) ∈ f (Rad (M)) ⊆ Rad (N) by [8, 21.6]. Hence Ker g ⊆ Rad (N). Theorem 2.2. An epimorphism f : M → N is a generalized cover if and only if for every homomorphism h : L → M such that f h : L → N is epic, h (L) is a weak generalized supplement of Ker f . Proof. (⇒) Let f : M → N be a generalized cover and let m ∈ M . Since f h is epic there exists l ∈ L such that f (m) = (f h) (l). Then m − h (l) ∈ Ker f and hence m ∈ h (L) + Ker f , which means that M = Ker f + h (L). By assumption, Ker f ⋂ h (L) ⊆ Rad (M) and so h (L) is a weak generalized supplement of Ker f . (⇐) It is clear that 1Mf = f is epic, for the identity homomorphism 1M : M → M . By the hypothesis, 1M (M) = M is a weak generalized supplement of Ker f , that is, Ker f ⊆ Rad (M). Hence f : M → N is a generalized cover. Proposition 2.3. Any homomorphic image of a WGS-module is a WGS- module. Proof. Let f : M → N be a homomorphism and M be a WGS−module. Suppose that U is a submodule of f (M). Then f−1 (U) is a submodule of M . Since M is a WGS-module, f−1 (U) has a weak generalized supplement V in M, i.e. f−1 (U) + V = M and f−1 (U) ⋂ V ⊆ Rad (M). Then f ( f−1 (U) ) + f (V ) = f (M). It follows that U + f (V ) = f (M). Note that U ⋂ f (V ) = f ( f−1 (U) ⋂ V ) ⊆ f (Rad (M)) ⊆ Rad (f(M)) by [8, 23.2]. Hence f (M) is a WGS-module. 3. Generalized ⊕-supplemented modules Recall from [6] that a module M is called ⊕-supplemented if every sub- module of M has a supplement that is a direct summand of M . Clearly ⊕-supplemented modules are supplemented. In this section, we define the concept of generalized ⊕-supplemented modules, which is adapted from Xue’s generalized supplemented modules, and give the properties of these modules. H. Çalı ş ıcı , E. Türkmen 13 Definition 3.1. A module M is called generalized ⊕-supplemented if every submodule of M has a generalized supplement that is a direct summand of M . Clearly ⊕-supplemented modules are generalized ⊕-supplemented. Also, finitely generated generalized ⊕-supplemented modules are ⊕-supp- lemented by [8, 19.3], but it is not generally true that every generalized ⊕- supplemented module is ⊕- supplemented. Let R be a non-local dedekind domain with quotient field K. Then the module K is generalized ⊕- supplemented, but it is not ⊕-supplemented. If K is ⊕-supplemented, R is a local ring by [10]. This is a contradiction by assumption. Later we shall give other examples of such modules (see Example 3.11). To prove that a finite direct sum of generalized ⊕-supplemented mod- ules is generalized ⊕-supplemented, we use the following standard lemma (see [8, 41.2]). Lemma 3.2. Let N and K be submodules of a module M such that N+K has a generalized supplement X in M and N ⋂ (K +X) has a generalized supplement Y in N . Then X + Y is a generalized supplement of K in M . Proof. Let X be a generalized supplement of N +K in M . Then M = (N +K) + X and (N +K) ⋂ X ⊆ Rad (X). Since N ⋂ (K +X) has a generalized supplement Y in N , we have N = N ⋂ (K +X) + Y and (K +X) ⋂ Y ⊆ Rad (Y ). Then M = N +K +X = [ N ⋂ (K +X) + Y ] +K +X = K + (X + Y ) and K ⋂ (X + Y ) ≤ X ⋂ (K + Y ) + Y ⋂ (K +X) ≤ X ⋂ (K +N) + Y ⋂ (K +X) ≤ Rad (X) + Rad (Y ) ≤ Rad (X + Y ) . Hence X + Y is a generalized supplement of K in M . Theorem 3.3. For any ring R, any finite direct sum of generalized ⊕- supplemented R-modules is generalized ⊕-supplemented. Proof. Let n be any positive integer and Mi (1 ≤ i ≤ n) be any finite collection of generalized ⊕-supplemented R-modules. Let M = M1⊕M2⊕ ...⊕Mn. Suppose that n = 2, that is, M = M1 ⊕ M2. Let K be any sub- module of M . Then M = M1 + M2 + K and so M1 + M2 + K has a generalized supplement 0 in M . Since M1 is generalized ⊕- supplemented, 14 Generalized ⊕-supplemented modules M1 ⋂ (M2 +K) has a generalized supplement X in M1 such that X is a direct summand of M1. By Lemma 3.2, X is a generalized supplement of M2 +K in M . Since M2 is generalized ⊕-supplemented, M2 ⋂ (K +X) has a generalized supplement Y in M2 such that Y is a direct summand of M2. Again applying Lemma 3.2, we have that X + Y is a generalized supplement of K in M . Since X is a direct summand of M1 and Y is a direct summand of M2, it follows that X ⊕ Y is a direct summand of M . The proof is completed by induction on n. We prove the following proposition, which is a modified form of Propo- sition 2.5 in [3]. We need the following lemma. Lemma 3.4. Let M be a module and N be a submodule of M . If U is a generalized supplement of N in M , then U+L L is a generalized supplement of N L in M L for every submodule L of N . Proof. By the hypothesis, M = N + U and U ⋂ N ⊆ Rad (U). Hence M L = N L + U+L L for every submodule L of N . Consider that the natural epimorphism φ : N → N L . Then by [8, p. 191], φ (Rad (U)) ⊆ Rad ( U+L L ) . Since U ⋂ N ⊆ Rad (U) it follows that N L ⋂ U + L L = L+ (N ⋂ U) L = = φ ( N ⋂ U ) ⊆ φ (Rad (U)) ⊆ Rad ( U + L L ) . Hence U+L L is a generalized supplement of N L in M L . Proposition 3.5. Let M be a nonzero generalized ⊕-supplemented R- module and let U be a submodule of M such that f (U) ≤ U for each f ∈ EndR (M). Then (1) The factor module M U is generalized ⊕-supplemented. (2) If, moreover, U is a direct summand of M , then U is also generalized ⊕-supplemented. Proof. (1) Let L U be any submodule of M U . Since M is generalized ⊕- supplemented, there exist submodules N and N ′ of M such that M = L + N , L ⋂ N ⊆ Rad (N) and M = N ⊕ N ′. By Lemma 3.4, N+U U is a generalized supplement of L U in M U . Since f (U) ≤ U for each f ∈ EndR (M), it follows from [3, Lemma 2.4] that U = (U ⋂ N)⊕ (U ⋂ N ′). Hence (N + U) ⋂ (N ′ + U) ≤ U and so N+U U ⋂ N ′+U U = 0, i.e. N+U U is a direct summand of M U . Thus M U is generalized ⊕-supplemented. H. Çalı ş ıcı , E. Türkmen 15 (2) Let U be a direct summand of M and let X be a submodule of U . Since M is generalized ⊕-supplemented, there exist submodules Y and Y ′ of M such that M = X + Y , X ⋂ Y ⊆ Rad (Y ) and M = Y ⊕ Y ′. Hence U = X + (U ⋂ Y ). Again applying [3, Lemma 2.4], we have that U = (U ⋂ Y ) ⊕ (U ⋂ Y ′). Now we show that X ⋂ (U ⋂ Y ) = X ⋂ Y ⊆ Rad (U ⋂ Y ). Let m be any element of X ⋂ Y . Then m ∈ Rad (Y ) and so Rm is small in Y . Since U is a direct summand of M , by [8, 19.3], Rm is small in U . Again by [8, 19.3], Rm is also small in U ⋂ Y because U ⋂ Y is direct summand of U . Hence m ∈ Rad (U ⋂ Y ). Consequently, U is generalized ⊕-supplemented. Corollary 3.6. Let M be a nonzero generalized ⊕-supplemented module. If Rad (M) is a direct summand of M , then Rad (M) is also generalized ⊕-supplemented. For a positive integer n, the modules Mi (1 ≤ i ≤ n) are called rela- tively projective if Mi is Mj-projective for all 1 ≤ i 6= j ≤ n. Theorem 3.7. Let Mi (1 ≤ i ≤ n) be any finite collection of relatively projective modules and let M = M1⊕M2⊕...⊕Mn. Then M is generalized ⊕-supplemented module if and only if Mi is generalized ⊕-supplemented for each 1 ≤ i ≤ n. Proof. (⇐) It follows from Theorem 3.3. (⇒) Clearly, it suffices to prove that M1 is generalized ⊕-supplemented. Let U be any submodule of M1. Since M is generalized ⊕-supplemented, there exist submodules V and V ′ of M such that M = U + V , U ⋂ V ⊆ Rad (V ) and M = V ⊕ V ′. By [6, Lemma 4.47], there exists a submodule V1 of V such that M = M1⊕V1. Then V = (M1 ⋂ V )⊕V1 and so M1 ⋂ V is a direct summand of M1. Now U ⋂ (M1 ⋂ V ) = U ⋂ V ⊆ Rad (V ) and thus U ⋂ V ⊆ Rad (M1 ⋂ V ) because M1 ⋂ V is a direct summand of V . Hence M1 is generalized ⊕-supplemented. Let R be a ring and M be an R-module. We consider the following condition. (D3) If M1 and M2 are direct summands of M with M = M1 +M2, then M1 ⋂ M2 is also a direct summand of M (see [6, p. 57]). Proposition 3.8. Let M be a generalized ⊕-supplemented module with (D3). Then every direct summand of M is generalized ⊕-supplemented. Proof. Let N be a direct summand of M and U be a submodule of N . Then there exists a direct summand V of M such that M = U + V and 16 Generalized ⊕-supplemented modules U ⋂ V ⊆ Rad (V ). It follows that N = U + (N ⋂ V ). Since M has (D3) N ⋂ V is a direct summand of M and so it is also a direct summand of N . Note that U ⋂ (N ⋂ V ) = U ⋂ V ⊆ Rad (V ). Since N ⋂ V is a direct summand of M , it follows that U ⋂ V ⊆ Rad (N ⋂ V ). Hence N is generalized ⊕-supplemented. Proposition 3.9 (see [2, Proposition 2.10]). Let M be a ⊕-supplemented module. Then M = M1⊕M2, where M1 is a module with Rad (M1) small in M1 and M2 is a module with Rad (M2) = M2. We give an analogous characterization of this fact for generalized ⊕-supplemented modules. Proposition 3.10. Let M be a generalized ⊕-supplemented module. Then M = M1 ⊕M2, where M1 is a module with Rad (M1) = M1 ⋂ Rad (M) and M2 is a module with Rad (M2) = M2. Proof. Since M is generalized ⊕-supplemented, there exist submodules M1 and M2 of M such that M = Rad (M) + M1, Rad (M) ⋂ M1 ⊆ Rad (M1) and M = M1 ⊕ M2. Then Rad (M1) = M1 ⋂ Rad (M) and M = M1 ⊕ Rad (M2). It follows that M2 = Rad (M2). Now we give some examples of module, which is generalized ⊕-supple- mented, but not ⊕-supplemented. Example 3.11. Let M be a non-torsion Z-module with Rad(M) = M . It is clear that M = Rad(M) is a generalized supplement of every submodule of M . Hence M is generalized ⊕-supplemented, but M is not supplemented by [10]. Consider the Z-module M = Q⊕ Z pZ , for any prime p. Note that M has a unique maximal submodule, i.e. Rad(M) 6= M . By Theorem 3.3, M is generalized ⊕-supplemented. If M is ⊕-supplemented, then Q is supplemented. It is a contradiction by [10]. Theorem 3.12. Let M be a module with (D3). Then the following state- ments are equivalent. (1) M is generalized ⊕-supplemented. (2) Every direct summand of M is generalized ⊕-supplemented. (3) There exists decomposition M = M1 ⊕M2 such that M1 is semisim- ple and M2 is a generalized ⊕-supplemented module with Rad (M2) essential in M2. H. Çalı ş ıcı , E. Türkmen 17 (4) There exists a decomposition M = M1 ⊕ M2 of M such that M1 is a generalized ⊕-supplemented module and M2 is a module with Rad (M2) = M2. Proof. (1) ⇒ (2) It follows from Proposition 3.8. (2) ⇒ (3) By [7, Proposition 2.3], M = M1 ⊕ M2, where M1 is semisimple and M2 is a module with Rad (M2) essential in M2. By (2), M2 is a generalized ⊕-supplemented. (3) ⇒ (1) By Theorem 3.3, M is generalized ⊕-supplemented. (1) ⇒ (4) By Proposition 3.10, there exist submodules M1 and M2 of M such that M = M1 ⊕M2 and Rad (M2) = M2. Since M has (D3), by Proposition 3.8, M1 is generalized ⊕-supplemented. (4) ⇒ (1) Since Rad (M2) = M2, M2 is generalized ⊕-supplemented. By (4) and Theorem 3.3, M is generalized ⊕-supplemented. A ring R is semiperfect if R Rad(R) is semisimple and idempotents can be lifted modulo Rad(R). It is known that a ring R is semiperfect if and only if every simple left R-module has a projective cover (see [8, 42.6]). Therefore it is shown in [4, Theorem 2.1] that R is semiperfect if and only if every finitely generated free R-module is ⊕-supplemented. Remark 3.13. For a ring R if every finitely generated free R-module is generalized ⊕-supplemented, then R is semiperfect. If RR is generalized ⊕-supplemented, RR is ⊕-supplemented because RR is a finitely generated R-module. It follows from [4, Theorem 2.1] that R is semiperfect. References [1] G. Azumaya, A characterization of semiperfect rings and modules, in “Ring Theory” edited by S. K. Jain and S. T. Rizvi, Proc. Biennial Ohio- Denison Conf., May 1992,World Scientific Publ., Singapore, 1993, pp. 28-40. [2] A. Harmancı, D. Keskin and P. F. Smith, On ⊕-supplemented modules, Acta Math. Hungar. 83 (1-2) (1999), 161-169. [3] A. Idelhadj and R. Tribak, On some properties of ⊕-supplemented modules, Int. J. Math. Math. Sci., 2003, (69) (2003), 4373-4387. [4] D. Keskin, P. F. Smith and W. Xue, Rings whose modules are ⊕-supplemented, J. Algebra 218 (1999), 161-169. [5] C. Lomp, Semilocal modules and rings, Comm. Algebra 27 (4) (1999), 1921-1935. [6] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, London Math. Soc. LNS 147 Cambridge Univ. Press (Cambridge, 1990). [7] Y. Wang and N. Ding, Generalized supplemented modules, Taiwan J. Math. 10 (6) (2006), 1589-1601. [8] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach (Philadelphia, 1991). 18 Generalized ⊕-supplemented modules [9] W. Xue, Characterizations of semiperfect and perfect rings, Publ. Mat. 40 (1996), 115-125. [10] H. Zöschinger, Komplementierte moduln über dedekindringen, J. Algebra 29(1974), 42-56. Contact information Hamza Çalışıcı Department of Mathematics, Faculty of Ed- ucation, Sakarya University, 54300, Sakarya, TURKEY E-Mail: hcalisici@sakarya.edu.tr Ergül Türkmen Department of Mathematics, Faculty of Arts and Science, Ondokuz Mayıs University, 55139, Samsun, TURKEY E-Mail: ergulturkmen@hotmail.com Received by the editors: 14.02.2010 and in final form 03.03.2011.

Generalized ⊕-supplemented modules

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