$\mathbb{Z}$ is a commutative ring and an ordered ring, the published construction being an instance of the general definitions

Let $\mathbb{Z}$ be the integers (The integers as equivalence classes of pairs of naturals) with the operations of Arithmetic on the integers and the order of Order on the integers. Then:

1. $(\mathbb{Z}, +, \cdot, 0, 1)$ is a ring in the sense of Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides, and it is commutative (Commutative ring);
2. $\mathbb{Z}$ with the order $\le$ is an ordered ring in the sense of Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication;
3. its positive cone $P = \{\, x \in \mathbb{Z} : 0 < x \,\}$ satisfies trichotomy and closure in the sense of The order presentation and the positive-cone presentation of an ordered ring determine each other: $P = \{\, x : 0 < x \,\}$ satisfies trichotomy and closure, and $a < b :\iff b - a \in P$ recovers the order, and the order it induces is the published order of Order on the integers.

The point of the example is that nothing is being built: the published The integers form a commutative ring and The integers form a totally ordered ring were proved before rings were defined and used the words "commutative ring" and "totally ordered ring" informally. This item records that those words, as used there, mean exactly what Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides and Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication mean, so that a statement about rings may be applied to $\mathbb{Z}$ without translation.

• The multiplicative axiom matches on the nose. The integers form a totally ordered ring proves the strict statement "$0 < x$ and $0 < y$ imply $0 < xy$", which is exactly axiom (OR2) of Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication as that definition states it. That is why the step establishing claim 2 above is a citation and not an argument, and it is the reason Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication adopts the strict form.

• $\mathbb{Z}$ is not a field, and it is an integral domain; both are recorded separately in $\mathbb{Z}$ is an integral domain of characteristic $0$ whose group of units is $\{1,-1\}$, so it is not a field: $2$ is nonzero and not invertible, for which this item supplies the ring structure.

• A later page that needs "$\mathbb{Z}$ is a ring" on its own spine must re-derive it there rather than cite this item, since examples pages are leaves in the reading order. The derivation is the three lines above.