# On the nilpotent residuals of all subalgebras of Lie algebras

• Volume: 68, Issue: 3, page 817-828
• ISSN: 0011-4642

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## Abstract

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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$, there exists a smallest ideal $I$ of $L$ such that $L/I\in 𝒩$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by ${L}^{𝒩}$. In this paper, we define the subalgebra $S\left(L\right)={\bigcap }_{H\le L}{I}_{L}\left({H}^{𝒩}\right)$. Set ${S}_{0}\left(L\right)=0$. Define ${S}_{i+1}\left(L\right)/{S}_{i}\left(L\right)=S\left(L/{S}_{i}\left(L\right)\right)$ for $i\ge 1$. By ${S}_{\infty }\left(L\right)$ denote the terminal term of the ascending series. It is proved that $L={S}_{\infty }\left(L\right)$ if and only if ${L}^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra $L$ with $S\left(L\right)=L$.

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