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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
It is also shown that for a group algebra K[G], if every proper cyclic right K[G]-module is Artinian and K-finite dimensional, then K[G] is Noetherian.
📌 3 Key Takeaways
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Objective & Hypothesis:Non-Artinian algebras over which proper cyclic right modules are Artinian must be right Ore domains.
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Key Findings & Data:It is shown that if R is a PI-ring whose proper cyclic right R-modules are Artinian, then R is right Noetherian.
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Conclusion & Impact:It is also shown that for a group algebra K[G], if every proper cyclic right K[G]-module is Artinian and K-finite dimensional, then K[G] is Noetherian.
🔬Design:Article
📍Scope:Comprehensive Sample
⚙️Variables:group algebra · proper cyclic · Artinian ring
Abstract
Non-Artinian algebras over which proper cyclic right modules are Artinian must be right Ore domains. It is shown that if R is a PI-ring whose proper cyclic right R-modules are Artinian, then R is right Noetherian. In particular, if G is a solvable group and each proper cyclic right K[G]-module is Artinian, then the group algebra K[G] is Noetherian. It is also shown that for a group algebra K[G], if every proper cyclic right K[G]-module is Artinian and K-finite dimensional, then K[G] is Noetherian.
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Cite This Article
Alahmadi, A. N. (2013). PI-RINGS WITH ARTINIAN PROPER CYCLICS ARE NOETHERIAN. International Electronic Journal of Algebra, Volume: 13 Issue: 13. https://ajindex.com/articles/1588224