AI 中文总结
研究将特征\(p\)环的结果扩展到含收缩自同态的环，利用\(\text{Hom}_R(^{\phi}R,-)\)得到正则环新特征，还研究\((-) \otimes_R {^{\phi}}R\)在局部上同调模与内射模上的行为。
AI 中文摘要
这项工作将先前为特征\(p\)的环建立的几个基本结果扩展到更广泛的允许收缩自同态的环类。具体而言，设\(\phi: R \to R\)为这样的自同态，并用\(^{\phi}R\)表示赋予由\(\phi\)诱导的左\(R -\)模结构的环\(R\)。我们利用函子\(\text{Hom}_R(^{\phi}R,-)\)得到正则环的新特征。此外，我们研究函子\((-) \otimes_R {^{\phi}}R\)在局部上同调模和内射模上的行为。
英文摘要
This work extends several fundamental results previously established for rings of characteristic $p$ to the broader class of rings that admit a contracting endomorphism. Specifically, let $ϕ: R \to R$ be such an endomorphism, and denote by $^ϕR$ the ring $R$ endowed with the left $R$-module structure induced by $ϕ$. We employ the functor $\text{Hom}_R(^ϕR,-)$ to derive new characterizations of regular rings. Furthermore, we examine the behavior of the functor $(-) \otimes_R {^ϕ}R$ on local cohomology modules and injective modules.
Comments17 pages
