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

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

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lobeflepnoumni

($$R,+$$,.) 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