In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and \(a\in R\) is a pure ad-nilpotent element of R of index n with R free of t and \(\left( {\begin{array}{c}n\\ t\end{array}}\right) \)-torsion for \(t=[\frac{n+1}{2}]\), then n is odd and there exists \(\lambda \in C(R)\) such that \(a-\lambda \) is nilpotent of index t. If R is a semiprime ring with involution \(*\) and a is a pure ad-nilpotent element of \({{\,\mathrm{Skew}\,}}(R,*)\) free of t and \(\left( {\begin{array}{c}n\\ t\end{array}}\right) \)-torsion for \(t=[\frac{n+1}{2}]\), then either a is an ad-nilpotent element of R of the same index n (this may occur if \(n\equiv 1,3 \,(\text {mod } 4)\)) or R is a nilpotent element of R of index \(t+1\), and R satisfies a nontrivial GPI (this may occur if \(n\equiv 0,3 \,(\text {mod } 4)\)). The case \(n\equiv 2 \,(\text {mod } 4)\) is not possible.

在本文中，我们研究了Lie代数中由无二阶半素联环R引起的幂幂子。为了控制R的扭转，我们引入了一个更严格的ad-幂零元素概念，即纯ad-nilpotent元素，这是唯一的技术条件，因为每个ad-nilpotent元素都可以表示为R的正交和。指数递减的纯无幂次元素。这使我们在设置环R内的扭转时更加精确，以描述其辅助力元素。如果R是一个半素环并且\（a \ in R \）是索引为n的R的纯ad-幂等元素，且其中R释放的吨和\（\左（{\开始{阵列} {C} n的\\吨\ {端阵列}} \右）\） -torsion为\（T = [\压裂{N + 1} {2 }] \），则n为奇数，并且存在\（\ lambda \ in C（R）\），使得\（a- \ lambda \）是索引t的幂等。如果R是具有对合\（* \）的半素环，而a是\（{{\，\ mathrm {Skew} \，}}（R，*）\）的纯ad幂零元素，且不包含t和\ （\ left（{\ begin {array} {c} n \\ t \ end {array}} \ right）\） - \（t = [\ frac {n + 1} {2}] \\的扭曲），），然后要么一是一个广告幂零元件ř相同索引的Ñ（这可能在发生\（N \当量1,3- \，（\ {文本MOD} 4）\） ）或- [R是幂零元件ř索引的\ （t + 1 \），并且R满足非平凡的GPI（如果\（n \ equiv 0,3 \，（\ text {mod} 4）\）可能会发生）。的情况下\（N \当量2 \，（\ {文本MOD} 4）\）是不可能的。