发表机构
Mälardalen University（马尔默达伦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入单侧理想的整性与完全整性概念及相应根构造，建立一般研究框架，并将Jacobson根相关经典结果推广至单侧情形，证明左Artin环与代数代数中左理想的Jacobson根整性及分次性，推广了Amitsur定理。
AI 中文摘要
我们引入了关于单侧理想的整性和完全整性的概念，以及相关的根构造，并为其研究建立了一个一般框架。作为应用，我们将若干关于Jacobson根的经典结果推广到单侧情形。特别地，我们证明了在左Artin环和代数代数中，左理想的Jacobson根关于该理想是整的，从而得到了Amitsur定理的一个单侧类比。我们还证明了分次左理想的Jacobson根是分次的，并由此推导出Amitsur关于多项式环的Jacobson根定理的一个相应推广。
英文摘要
We introduce notions of integrality and full integrality over one-sided ideals, together with associated radical constructions, and develop a general framework for their study. As applications, we extend several classical results concerning Jacobson radicals to the one-sided setting. In particular, we prove that the Jacobson radical of a left ideal is integral over that ideal in left artinian rings and in algebraic algebras, obtaining a one-sided analogue of a theorem of Amitsur. We also show that Jacobson radicals of graded left ideals are graded and derive a corresponding extension of Amitsur's theorem on Jacobson radicals of polynomial rings.
