δ-M-Small and δ-Harada Modules
Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective gene...
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| Published in | Communications in algebra Vol. 36; no. 2; pp. 423 - 433 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Taylor & Francis Group
01.02.2008
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| Abstract | Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ
M
-lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained. |
|---|---|
| AbstractList | Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a δ-M-small module if N≅ K, K is δ-M-small in L for some K, L ∈ σ[M]. In this article, we prove that if M is a finitely generated self-projective generator in σ[M], then M is a Noetherian QF-module if and only if every module in σ[M] is a direct sum of a projective module in σ[M] and a δ-M-small module. As a generalization of a Harada module, a module M is called a δ-Harada module if every injective module in σ[M] is δ
M
-lifting. Some properties of δ-Harada modules are investigated and a characterization of a Harada module is also obtained. Let M be a right R-module and Nepsilonsigma[M]. A submodule K of N is called delta-M-small if, whenever N=K+X with N/X M-singular, we have N=X. N is called a delta-M-small module if N{congruent to} K, K is delta-M-small in L for some K, Lepsilonsigma[M]. In this article, we prove that if M is a finitely generated self-projective generator in sigma[M], then M is a Noetherian QF-module if and only if every module in sigma[M] is a direct sum of a projective module in sigma[M] and a delta-M-small module. As a generalization of a Harada module, a module M is called a delta-Harada module if every injective module in sigma[M] is deltaM-lifting. Some properties of delta-Harada modules are investigated and a characterization of a Harada module is also obtained. |
| Author | Özcan, A. Çiğdem Koşan, M. Tamer |
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| References | Anderson F. W. (CIT0002) 1974 Koşan M. T. (CIT0007) 2007; 14 Page S. S. (CIT0012) 1994; 46 Al-Khazzi I. (CIT0001) 1991; 19 Goodearl K. R. (CIT0004) 1976 Dung N. V. (CIT0003) 1994 Jayaraman M. (CIT0006) 2000; 28 Zhou Y. (CIT0015) 2000; 7 Oshiro K. (CIT0010) 1984; 13 Rayar M. (CIT0013) 1982; 39 CIT0005 Wisbauer R. (CIT0014) 1991 Mohamed S. H. (CIT0008) 1990 Oshiro K. (CIT0009) 1984; 13 (CIT0011) 2002; 51 |
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| Snippet | Let M be a right R-module and N ∈ σ[M]. A submodule K of N is called δ-M-small if, whenever N = K + X with N/X M-singular, we have N = X. N is called a... Let M be a right R-module and Nepsilonsigma[M]. A submodule K of N is called delta-M-small if, whenever N=K+X with N/X M-singular, we have N=X. N is called a... |
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| SubjectTerms | Harada module and ring Injective module Lifting module Noetherian QF-module Small module |
| Title | δ-M-Small and δ-Harada Modules |
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