Right Identity Element is Idempotent

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Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $e_R \in S$ be a right identity with respect to $\circ$.


Then $e_R$ is idempotent under $\circ$.


Proof

By the definition of a right identity:

$\forall x \in S: x \circ e_R = x$

Thus in particular:

$e_R \circ e_R = e_R$

Therefore $e_R$ is idempotent under $\circ$.

$\blacksquare$


Also see


Sources