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Every element of some ring $(R,+,*)$ is such that $a*a=a.$ This ring

1. is commutative
2. is non-commutative
3. may or may not be commutative
4. none

Rings aren't there in syllabus ,right ?
why may or maynot commutative?
No, its actually commutative. Was a typo in key, corrected now.

Consider

$(x+y).(x+y) = x+y$

$\implies (x+y).x + (x+y).y = x+y$

$\implies x.x + y.x + x.y + y.y = x+y$

$\implies x + y.x + x.y + y = x + y$

$\implies x+ y + y.x + x.y = x+ y$

$\implies x.y = - y.x \to (1)$

Now, we have $a.a = a \\ \implies (a+a) . (a+a) = a+a \\ \implies a.a + a.a + a.a + a.a = a + a \\ \implies a.a + a.a = 0 \\ \implies a + a = 0 \\\implies a = -a \to (2)$

From (1) and (2),

$x.y = y.x, \forall x, y$

So, the given ring is commutative.
by

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