The aim of this paper is to investigate different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) of the noncommutative rings known as skew Poincaré–Birkhoff–Witt extensions. We characterize minimal prime ideals of these rings and prove that the Köthe’s conjecture holds for these extensions. Finally, we establish the transfer of several ring-theoretical properties (reduced, symmetric, reversible, 2-primal) from the coefficients ring of a skew PBW extension to the extension itself.

中文翻译：

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偏斜PBW扩展的自由基和Köthe猜想

本文旨在研究非交换环的不同自由基（Wedderburn自由基，下零自由基，Levitzky自由基，上零自由基，所有零能元素的集合，所有零理想理想的总和），称为偏斜庞加莱–伯克霍夫–Witt扩展。我们刻画了这些环的最小理想理想，并证明了Köthe的猜想对于这些扩展是成立的。最后，我们建立了从偏斜PBW扩展的系数环到扩展本身的几个环理论性质（还原的，对称的，可逆的，2阶的）的传递。