Question

2.23. Let R be a commutative ring. Suppose that P is a subset of R with the following properties: P1: For any element a ∈ R, one and only one of the following holds: a = 0, a ∈ P, or −a ∈ P. P2: P is closed under addition and multiplication. Define an order relation < on R by saying that for a, b ∈ R, a < b if b − a ∈ P. Show that < satisfies Axioms O1–O4 .

Axiom O1 (Trichotomy). For any two numbers, a and b, one and only one of the following statements is true: a < b, a = b, or b < a.

Axiom O2 (Transitivity). For any three numbers, a, b, and c, if a < b and b < c, then a < c.

Axiom O3 (Addition for Inequalities). For any three numbers, a, b, and c, if a < b then a + c < b + c.

Axiom O4 (Multiplication for Inequalities). For any three numbers, a, b, and c, if a < b and 0 < c, then ac < bc.

Answer 1