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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
If R is semicommutative and weak α-rigid, it is proven that the ring R[x] and the ring (R[x]/(xn)) where (xn) is the ideal generated by xn and n is a positive integer, are weak α-skew Armendariz.
📌 3 Key Takeaways
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Objective & Hypothesis:For a ring endomorphism α, we introduce weak α-rigid and weak α-skew Armendariz rings which are a generalization of α-rigid rings, and investigated their properties.
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Key Findings & Data:Moreover, we prove that a ring R is weak α-rigid if and only if for any n, the n by n upper triangular matrix ring Tn(R) is weak α-rigid.
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Conclusion & Impact:If R is semicommutative and weak α-rigid, it is proven that the ring R[x] and the ring (R[x]/(xn)) where (xn) is the ideal generated by xn and n is a positive integer, are weak α-skew Armendariz.
🔬Design:Article
📍Scope:Comprehensive Sample
⚙️Variables:weak α-rigid · skew Armendariz ring · extension
Abstract
For a ring endomorphism α, we introduce weak α-rigid and weak α-skew Armendariz rings which are a generalization of α-rigid rings, and investigated their properties. Moreover, we prove that a ring R is weak α-rigid if and only if for any n, the n by n upper triangular matrix ring Tn(R) is weak α-rigid. If R is semicommutative and weak α-rigid, it is proven that the ring R[x] and the ring (R[x]/(xn)) where (xn) is the ideal generated by xn and n is a positive integer, are weak α-skew Armendariz.
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Cite This Article
Ouyang, L. (2008). EXTENSIONS OF GENERALIZED α-RIGID RINGS. International Electronic Journal of Algebra, Volume: 3 Issue: 3. https://ajindex.com/articles/1588357