Direct sums of dual continuous modules

Mohamed, Saad H.; Müller, Bruno J.

doi:10.1007/bf01262041

Cited by 5 publications

(7 citation statements)

References 8 publications

(4 reference statements)

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“…We also give some partial converses of this theorem, which extend analogous results for ^-continuous modules due to Mohamed and Muller [11,12].…”

Section: If M X M Is Qd-continuous Then M Is Quasi-projectivementioning

confidence: 61%

“…As a consequence we get the following result which extends [12 [7] Quasi-dual-continuous modules 293…”

Section: Let M Be a Qd-continuous Module And B A D-complement Of A Smentioning

confidence: 79%

“…In [11] and [13] Mohamed, Muller and Singh established a decomposition theorem for J-continuous modules. Then dual continuous modules, and modules satisfying (D x ) only, were further studied by Abdul-Karim, Mohamed, Muller and Singh in [8], [9], [12], [16], [17]. Jeremy [5] defined the concept of quasi-continous modules.…”

mentioning

confidence: 99%

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Quasi-dual-continuous modules

Mohamed

Müller

Singh³

1985

J Aust Math Soc A

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Quasi-dual-continuous modules, which generalize the concept of dual-continuous modules, are studied Mohamed, Muller and Singh had obtained some decomposition theorems and their partial canvwsfts, for dual-continuous modules. It is shown that these results can be extended to quasi-dual-continuous modules. Further, a short proof of a decomposition theorem for quasi-dual-continuous modules established recently by Oshiro is given. Some more structure theorems for such modules are established. Finally, quasi-dual-continuous covers are studied, and duals for results of Muller and Rizvi are derived.

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“…We also give some partial converses of this theorem, which extend analogous results for ^-continuous modules due to Mohamed and Muller [11,12].…”

Section: If M X M Is Qd-continuous Then M Is Quasi-projectivementioning

confidence: 61%

“…As a consequence we get the following result which extends [12 [7] Quasi-dual-continuous modules 293…”

Section: Let M Be a Qd-continuous Module And B A D-complement Of A Smentioning

confidence: 79%

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Quasi-dual-continuous modules

Mohamed

Müller

Singh³

1985

J Aust Math Soc A

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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. A module M is called a completely ⊕-supplemented (see [5]) (or satisfies (D + 11 ) in our terminology, see [6,7]) if every direct summand of M is ⊕-supplemented.…”

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

confidence: 99%

“…Our next objective is to prove that over a commutative ring, if M is a finitely generated ⊕-supplemented module with corank(M) = 3, then M is a direct sum of local modules. We first prove the following generalization of [11,Lemma 2.3].…”

Section: Lemma 24 Let M Be a Nonzero Module And Let U Be A Submodulmentioning

confidence: 99%

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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Skew-injective modules

Tuganbaev¹

1999

J Math Sci

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Direct sums of dual continuous modules

Cited by 5 publications

References 8 publications

Quasi-dual-continuous modules

Quasi-dual-continuous modules

On some properties of ⊕‐supplemented modules

Skew-injective modules

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