Identity Mapping is Ordered Ring Automorphism

Theorem

Let $\struct {S, +, \circ, \preceq}$ be an ordered ring.

Then the identity mapping $I_S: S \to S$ is an ordered ring automorphism.

Proof

We have that:

an identity mapping is an order isomorphism

an identity mapping is a group automorphism

an identity mapping is a semigroup automorphism

Hence the result by definition of ordered ring automorphism.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers