群胚分次环的Hopkins-Levitzki型定理
Hopkins-Levitzki Type Theorems for Groupoid Graded Rings
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中文总结 AI 辅助
该研究针对群胚分次环，引入分次理想的左/右逐对象幂零性概念，推广Hopkins-Levitzki定理，证明三类Γ₀-Artinian环为Γ₀-Noetherian环，并用反例说明相关结论。
中文摘要 AI 辅助
我们继续研究对象单位群胚分次环的基础理论，本工作特别关注分次Jacobson根的幂零性条件。我们引入分次理想的左/右逐对象幂零性概念，证明该条件适用于得到Hopkins-Levitzki定理的分次推广。尽管该条件非对称，我们表明其双侧版本适合定义gr-半准素环。已知单侧Γ₀-Artinian环未必是Γ₀-Noetherian环，但利用我们的工具可证明：单侧gr-遗传Γ₀-Artinian环、双侧Γ₀-Artinian环及d-有限生成的单侧Γ₀-Artinian环均为Γ₀-Noetherian环。不过，第一类未必是gr-半准素环，而另外两类总是如此。我们用若干（反）例说明结果，尤其涉及分次上三角矩阵。
英文摘要
We continue the study of the basic theory of object-unital groupoid graded rings. In this work, we are especially interested in nilpotency conditions on the graded Jacobson radical. We introduce the concept of left/right objectwise nilpotency of graded ideals, and prove that this condition is appropriate for obtaining graded generalizations of the Hopkins--Levitzki theorem. Although this condition is not symmetric, we show that its two-sided version is suitable for defining gr-semiprimary rings. It is known that one-sided $Γ_0$-artinian rings need not be $Γ_0$-noetherian, but using our tools we prove that one-sided gr-hereditary $Γ_0$-artinian rings, two-sided $Γ_0$-artinian rings, and $d$-finitely generated one-sided $Γ_0$-artinian rings are $Γ_0$-noetherian. However, the first class need not be gr-semiprimary, whereas the other two always are. We illustrate our results with several (counter)examples, especially involving graded upper triangular matrices.
发表机构
- University of São Paulo（圣保罗大学）
- Federal University of Mato Grosso do Sul（南马托格罗索联邦大学）
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