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3 | \cdots | ||
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If $1-ab$ is invertible for $a$, $b$ in a (noncommutative) ring then so is $1-ba$. Proof:
$$(1-ba)^{-1} = 1+ba +baba+... baba+\cdots = 1+b(1+ab+abab+..)a 1+b(1+ab+abab+\cdots)a = 1+b(1-ab)^{-1}a,$$ |
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2 | just latexified some things for completeness | ||
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If 1-ab $1-ab$ is invertible for a, b $a$, $b$ in a (noncommutative) ring then so is 1-ba. $1-ba$. Proof:
$(1-ba)^{-1} $(1-ba)^{-1} = 1+ba +baba+... = 1+b(1+ab+abab+..)a = 1+b(1-ab)^{-1}a$,1+b(1-ab)^{-1}a,$$ |
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1 | [made Community Wiki] | ||
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If 1-ab is invertible for a, b in a (noncommutative) ring then so is 1-ba. Proof:
$(1-ba)^{-1} = 1+ba +baba+... = 1+b(1+ab+abab+..)a = 1+b(1-ab)^{-1}a$, |
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