If a is an idempotent in a commutative ring, show that 1 - a is also an idempotent.

Annette Arroyo 2020-10-20 Answered

If a is an idempotent in a commutative ring, show that \(1 - a\) is also an idempotent.

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lobeflepnoumni
Answered 2020-10-21 Author has 25719 answers

(\(R,+\),.) commutative ring and \(\displaystyle{a}\in{R}\)
Such that a is idempotenr element
i.e \(\displaystyle{a}^{{2}}={a}\)
\(\displaystyle{\left({1}-{a}\right)}^{{2}}={\left({1}-{a}\right)}{\left({1}-{a}\right)}\)
\(\displaystyle={1}-{a}-{a}+{2}^{{2}}\)
\(\displaystyle={1}-{2}{a}+{a}{\left\lbrace{\sin{{c}}}{e}{a}^{{2}}={a}\right\rbrace}\)
\(\displaystyle={1}-{a}\)
\(\displaystyle\Rightarrow{\left({1}-{a}\right)}^{{2}}={1}-{a}\)
\(\displaystyle\Rightarrow\) \(1-a\) is idempotent element

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