Identity Element is Idempotent

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Theorem

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

Let $e \in S$ be an identity with respect to $\circ$.

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

Proof 1

By the definition of an identity element:

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

Thus in particular:

$e \circ e = e$

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

$\blacksquare$

Proof 2

Follows from Left Identity Element is Idempotent, Right Identity Element is Idempotent and definition of Two-Sided Identity.

$\blacksquare$

Also see

• Left Identity Element is Idempotent
• Right Identity Element is Idempotent

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.8)$