A dual notion of CS-modules generalization

Idelhadj, Abdelouahab; Tribak, Rachid

doi:10.1201/9780203903889.ch13

Cited by 5 publications

(7 citation statements)

References 6 publications

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“…By ( (Warfield Jr., 1970), Theorem 2), there exists a finitely presented indecomposable module M = R (n) /K which can not be generated by fewer than n elements. By ((Idelhadj and Tribak, 2000), Corollary 1), R (n) is ⊕-supplemented, and therefore ⊕-cocoatomically supplemented. However, M is not ⊕-cofinitely supplemented, so it is not ⊕-co-coatomically supplemented (see (Idelhadj and Tribak, 2000), Proposition 2 and (Wang and Sun, 2007), Example 2.1).…”

Section: Example 43 Let R Be a Commutative Local Ring Which Is Not Amentioning

confidence: 99%

“…By ((Idelhadj and Tribak, 2000), Corollary 1), R (n) is ⊕-supplemented, and therefore ⊕-cocoatomically supplemented. However, M is not ⊕-cofinitely supplemented, so it is not ⊕-co-coatomically supplemented (see (Idelhadj and Tribak, 2000), Proposition 2 and (Wang and Sun, 2007), Example 2.1).…”

Section: Example 43 Let R Be a Commutative Local Ring Which Is Not Amentioning

confidence: 99%

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Co-Coatomically Supplemented Modules

Alizade

Güngör

2017

Ukr Math J

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Section: Example 43 Let R Be a Commutative Local Ring Which Is Not Amentioning

confidence: 99%

Section: Example 43 Let R Be a Commutative Local Ring Which Is Not Amentioning

confidence: 99%

Co-Coatomically Supplemented Modules

Alizade

Güngör

2017

Ukr Math J

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“…In [17] Smith and Tercan investigate the following property which they called (C 11 ): every submodule of M has a complement which is a direct summand of M. So, it was natural to introduce a dual notion of (C 11 ) which we called (D 11 ) (see [6,7]). It turns out that modules satisfying (D 11 ) are exactly the ⊕-supplemented modules.…”

Section: Page 95] Mohamed and Müller Called A Module ⊕-Supplemented mentioning

confidence: 99%

“…(ii) If R is a commutative ring and M a finitely generated ⊕-supplemented module with corank(M) = 3, then M is completely ⊕-supplemented (see [6,Corollary 6] and Corollary 3.9).…”

Section: Remark 310 (I) Ifmentioning

confidence: 99%

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On some properties of ⊕‐supplemented modules

Idelhadj

Tribak

2003

International Journal of Mathematics and Mathematical Sciences

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A module M is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.2000 Mathematics Subject Classification: 16D50, 16D60, 13E05, 13E15, 16L60, 16P20, 16D80. Introduction.All rings considered in this paper will be associative with an identity element. Unless otherwise mentioned, all modules will be left unitary modules. Let R be a ring and M an R-module. Let A and P be submodules of M. The submodule P is called a supplement of A if it is minimal with respect to the property A + P = M. Any L ≤ M which is the supplement of an N ≤ M will be called a supplement submodule of M. If every submodule U of M has a supplement in M, we call M complemented. In [25, page 331], Zöschinger shows that over a discrete valuation ring R, every complemented R-module satisfies the following property (P ): every submodule has a supplement which is a direct summand. He also remarked in [25, page 333] that every module of the form M (R/a 1 ) × ··· × (R/a n ), where R is a commutative local ring and a i (1 ≤ i ≤ n) are ideals of R, satisfies (P ). In [12, page 95], Mohamed and Müller called a module ⊕-supplemented if it satisfies property (P ).On the other hand, let U and V be submodules of a module M. The submodule V is called a complement of U in M if V is maximal with respect to the property V ∩U = 0. In [17] Smith and Tercan investigate the following property which they called (C 11 ): every submodule of M has a complement which is a direct summand of M. So, it was natural to introduce a dual notion of (C 11 ) which we called (D 11 ) (see [6,7]). It turns out that modules satisfying (D 11 ) are exactly the ⊕-supplemented modules. A module M is called a completely ⊕-supplemented (see [5]) (or satisfies (D

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On $(D_{12})$-modules

Tütüncü¹,

Tribak²

2013

Rocky Mountain J. Math.

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A dual notion of CS-modules generalization

Cited by 5 publications

References 6 publications

Co-Coatomically Supplemented Modules

Co-Coatomically Supplemented Modules

On some properties of ⊕‐supplemented modules

On $(D_{12})$-modules

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