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
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.2$