JACOBSON’S LEMMA FOR DRAZIN INVERSES
JACOBSON’S LEMMA FOR DRAZIN INVERSES
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DOI：
10.1090/conm/609/12127
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
T. Lam;Pace P. Nielsen
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文献类型：
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作者：
T. Lam;Pace P. Nielsen
如果环元素α = 1−ab是Drazin可逆的，则β = 1−ba也是。我们证明了β的Drazin逆可以用一个简单的公式（推广“Jacobson引理”）用Drazin逆和α的谱幂等表示。将αa = a β和b α = β b的对易规则推广到Drazin逆上，证明α和β的谱幂等是同构的。在一个相关的方向上，我们也证明了Jacobson引理对任意环上的π-正则元和单位π-正则元成立。
If a ring element α = 1 − ab is Drazin invertible, so is β = 1 − ba. We show that the Drazin inverse of β is expressible by a simple formula (in generalization of “Jacobson’s Lemma”) in terms of the Drazin inverse and the spectral idempotent of α. The commutation rules αa = a β and b α = β b are extended to the Drazin inverses, and the spectral idempotents of α and β are shown to be isomorphic. In a related direction, we also prove that Jacobson’s Lemma holds for π-regular elements and unit π-regular elements in arbitrary rings.