$\mathbb{Q}$ and $\mathbb{R}$ are fields, hence commutative rings, integral domains and ordered rings, all of characteristic $0$

Let $F$ be either $\mathbb{Q}$ (The rationals form a field) or $\mathbb{R}$ (The reals form a field), with its published order (The rationals form a totally ordered field, The reals form a totally ordered field). Then:

1. $F$ is a commutative ring with $1 \ne 0$ and an integral domain (Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring, Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors);
2. $F$ with its order is an ordered ring (Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication), and the set $P = \{\, x \in F : 0 < x \,\}$ is a positive cone making $F$ an ordered field in the sense of Ordered field, whose induced order is the published one;
3. $\operatorname{char}(F) = 0$ (The characteristic of a ring: the least $n \ge 1$ with $n \cdot 1_R = 0$ when one exists, and $0$ otherwise).

• Two presentations of one order, reconciled here rather than assumed. The published The rationals form a totally ordered field and The reals form a totally ordered field state the order form; the published Ordered field states the positive-cone form. The step establishing claim 2 above passes between them using 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 that is the only reason Canonical naturals are positive and strictly increasing, which is stated for an ordered field, may be applied to $\mathbb{Q}$ and $\mathbb{R}$ here.

• The multiples agree with the canonical naturals. By [L6] the element $n \cdot 1_F$ appearing in The characteristic of a ring: the least $n \ge 1$ with $n \cdot 1_R = 0$ when one exists, and $0$ otherwise is the $\iota(n)$ of The canonical natural $\iota(n) = n \cdot 1_F$ of a field, so claim 3 is also the statement that $\iota$ never takes the value $0_F$ on $n \ge 1$ in these two fields. More is true and is quoted rather than proved here: Canonical naturals are positive and strictly increasing shows $\iota$ is strictly increasing on $n \ge 1$, hence injective.

• $\mathbb{Q}$ and $\mathbb{R}$ are domains for a reason stronger than necessary. They have no zero divisors because every nonzero element is invertible, not because of any cancellation argument; the same reasoning gives nothing about $\mathbb{Z}$, whose domain property is 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.