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arXiv · 1702.06271

Jacobson's lemma via Groebner-Shirshov bases

Xiangui Zhao
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Abstract

Let $R$ be a ring with identity $1$. Jacobson's lemma states that for any $a,b\in R$, if $1-ab$ is invertible then so is $1-ba$. Jacobson's lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Groebner-Shirshov basis theory to obtain the inverse of $1-ab$ in terms of $(1-ba)^{-1}$, assuming the latter exists.

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Xiangui Zhao. 2017-02-21. Jacobson's lemma via Groebner-Shirshov bases. https://arxiv.org/abs/1702.06271

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