Ojective Modules

Mohamed, Saad H.; Müller, Bruno J.

doi:10.1081/agb-120013218

Cited by 16 publications

(15 citation statements)

References 3 publications

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“…In this section we obtain a lattice theoretic analogue of some results from Mohamed and Müller, [12]. They studied the concept of ojectivity for direct summands of modules.…”

Section: Ojective Ideals In Modular Latticesmentioning

confidence: 91%

“…In the honor of Oshiro, in [12], Mohamed and Müller call an M -injective module an ojective module. They proved that mutual ojectivity is a necessary and sufficient condition for the direct sum of extending modules to be extending.…”

Section: Introductionmentioning

confidence: 99%

“…generalized relative injectivity) in the context of the lattice of ideals in a modular lattice with a certain condition. The key point for this translation is the following result from Mohamed and Müller [12], Theorem 7 which characterizes ojective modules in terms of lattices of submodules. (The word complement used in this theorem has the following meaning: C is a complement of B if C is maximal with the property C ∩ B = {0}.…”

Section: Introductionmentioning

confidence: 99%

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Ojective ideals in modular lattices

Nimbhorkar

Shroff

2015

Czech Math J

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“…In this section we obtain a lattice theoretic analogue of some results from Mohamed and Müller, [12]. They studied the concept of ojectivity for direct summands of modules.…”

Section: Ojective Ideals In Modular Latticesmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Ojective ideals in modular lattices

Nimbhorkar

Shroff

2015

Czech Math J

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“…They changed the phrase "generalized relative injective" into "relative objective." As we saw in Proposition 2.6, if M and N are CS and mutually relative generalized injective, then M is generalized N-injective for any direct summand M of M. However, in [11], it is shown that if M is generalized N-injective, then any direct summand M of M is generalized N-injective.…”

Section: -Injective and Essentiallymentioning

confidence: 96%

On Direct Sums of Extending Modules and Internal Exchange Property

Hanada¹,

Kuratomi

Oshiro

2002

Journal of Algebra

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An R-module M is called an extending module, and also called CS, if it satisfies the following full extending property: For any submodule X of M, there exists a direct summand X * of M, which is an essential extension of X. The concept of this module is a notable property of injective modules, quasi-injective modules, continuous modules, and quasi-continuous modules. In the early days of ring theory, this extending property appeared in Utumi [17] as a von Neumann regular ring R is right continuous if and only if R is an extending module as a right R-module. On the other hand, in [2][3][4], by a quite different method, Harada introduced several extending properties and, simultaneously, lifting properties which are mutually dual notions. These papers not only quickened the progress of the study of continuous and quasi-continuous modules due to Jeremy [8] but also much influenced ring and module theory. For background and applications of extending and lifting properties, the reader is referred to the texts of Harada

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“…In the second case, we have Next we give characterizations for different types of injectivity analogous to that given in [8] for ojective modules. First we need some lemmas.…”

Section: Mixed Injectivitymentioning

confidence: 99%

Mixed Injective Modules

Tütüncü¹,

Mohamed²,

Ertaş³

2010

Glasgow Math. J.

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Abstract. Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.2000 Mathematics Subject Classification. Primary 16D50, Secondary 16D80. B is almost A-injective if for any ϕ : A ≥ X −→ B, there exists a homomorphism ϕ 1 : A −→ B that extends ϕ (injectivity behaviour), or there exists a non-zero summand A 2 ≤ A and a homomorphism ϕ 2 : B −→ A 2 such that ϕ 2 ϕ = π A 2 | X , where π A 2 is the projection of A onto A 2 (which we refer to as opposite injectivity behaviour). We note that for an indecomposable module A, we have that B is almost A-injective if and only if for any ϕ : A ≥ X −→ B, there exists a homomorphism ϕ 1 : A −→ B or a homomorphism ϕ 2 : B −→ A such that the following diagrams commute:

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Ojective Modules

Cited by 16 publications

References 3 publications

Ojective ideals in modular lattices

Ojective ideals in modular lattices

On Direct Sums of Extending Modules and Internal Exchange Property

Mixed Injective Modules

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