On semi-artinian modules and injectivity conditions

Clark, John; Smith, Patrick F.

doi:10.1017/s0013091500022999

Cited by 8 publications

(3 citation statements)

References 26 publications

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“…Recall that a module $X$ is semiartinian if and only if every proper factor module of $X$ has a simple submodule (see [10, p. 182]). Dualizing this, we say that a module is seminoetherian in case every nonzero submodule has a maximal submodule (see [4]). THEOREM 5.…”

Section: $\Square $mentioning

confidence: 99%

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On second socles of finitely cogenerated injective modules

Koike¹

1998

Tsukuba J. Math.

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finitely cogenerated seminoetherian right R-module is of finite length (Theorem 5). Here, seminoetherian modules mean modules whose every nonzero submodule contains a maximal submodule. As a corollary, we obtain the theorem of Clark and Huynh (Corollary 7). Also we point out a condition for certain right perfect rings to have Morita duality (Corollary 10). In the last part of this note, we mention a dual of Theorem 5 (Theorem 13).

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Section: $\Square $mentioning

confidence: 99%

“…Conceming this, several results are shown recently. In [4,Corollary 4] the authors point out that the perfect condition can be weakened to semiartinian. Other authors approach to the problem above by investigating the condition that the lefi R-module $Soc_{2}(R)$ (or $J/J^{2}$ ) is finitely (countably) generated.…”

Section: 4])mentioning

confidence: 99%

On second socles of finitely cogenerated injective modules

Koike¹

1998

Tsukuba J. Math.

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“…Further, a ring R is said to be right semi-artinian if every non-zero right R-module has a non-zero socle. Semi-artinian rings and modules were investigated, for example, in [3] and [5].…”

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confidence: 99%

A Result on Semi-Artinian Rings

Dinh

Smith

2003

Proceedings of the Edinburgh Mathematical Society

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It was shown by Huynh and Rizvi that a ring R is semisimple artinian if and only if every continuous right R-module is injective. However, a characterization of rings, over which every finitely generated continuous right module is injective, has been left open. In this note we give a partial solution for this question. Namely, we show that for a right semi-artinian ring R, every finitely generated continuous right R-module is injective if and only if all simple right R-modules are injective. (p) Every finitely generated continuous right R-module is injective.(q) Every finitely generated CS right R-module is (quasi-) continuous.The structure of rings satisfying either (p) or (q) is unknown. However, such a ring need not be semisimple artinian. It is shown in [9] that any simple right and left SI ring with 63

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Rings Whose Modules Have Grade Zero

2005

Acta Math Sinica

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On semi-artinian modules and injectivity conditions

Cited by 8 publications

References 26 publications

On second socles of finitely cogenerated injective modules

On second socles of finitely cogenerated injective modules

A Result on Semi-Artinian Rings

Rings Whose Modules Have Grade Zero

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