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An element $x$ of ring $R$ is called idempotent if $x^2 = x$.

Which rings do not have idempotent elements other than $0$ and $1$?

Nontrivial commutative ring $R$

Nontrivial cancellative ring $R$

Ring $R$ with no nontrivial nilpotent elements

$M_2(\mathbb{R})$

Ring $R$ with $y \ne 0, 1$, and $y^2 = y^4$