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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 2

Semisimple-direct-injective modules

Adel Abyzov(Kazan (Volga Region) Federal University)0000-0002-9809-2091
Muhammet Tamer Koşan(Gazı University)0000-0002-5071-4568
Truong Cong Quynh(The University of Danang)0000-0002-0845-0175
Daniel Tapkin(Kazan (Volga Region) Federal University)0000-0003-0828-4397
Pages
—
Publication Date
2021-04-11
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We also show that $R$ is a right Noetherian right V-ring if and only if every right $R$-module has a semisimple-direct-injective envelope if and only if every right $R$-module has a semisimple-direct-injective cover.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The notion of simple-direct-injective modules which are a generalization of injective modules unifies $C2$ and $C3$-modules.
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Key Findings & Data:In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of $C2$, $C3$, SSP properties and simple-direct-injective modules.
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Conclusion & Impact:We also show that $R$ is a right Noetherian right V-ring if and only if every right $R$-module has a semisimple-direct-injective envelope if and only if every right $R$-module has a semisimple-direct-injective cover.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:$C2$-module · $C3$-module · SSP

Abstract

The notion of simple-direct-injective modules which are a generalization of injective modules unifies $C2$ and $C3$-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of $C2$, $C3$, SSP properties and simple-direct-injective modules. It is proved that a ring $R$ is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right $R$-module has the SSP and, for any family of simple injective right $R$-modules $\{S_i\}_{\mathcal{I}}$, $\oplus_{\mathcal{I}}S_i$ is injective. We also show that $R$ is a right Noetherian right V-ring if and only if every right $R$-module has a semisimple-direct-injective envelope if and only if every right $R$-module has a semisimple-direct-injective cover.

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Cite This Article

Abyzov, A., Koşan, M. T., Quynh, T. C., Tapkin, D. (2021). Semisimple-direct-injective modules. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 2. https://doi.org/10.15672/hujms.730907

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